삼각함수 적분표

1. 개요

삼각함수 적분표는 사인, 코사인, 탄젠트, 시컨트, 코시컨트, 코탄젠트 함수를 포함하는 삼각함수의 다양한 적분 공식들을 제공한다. 이 표는 일반적인 삼각함수, 거듭제곱 형태, x와 삼각함수의 곱, 여러 삼각함수의 곱, 분수 형태의 삼각함수, 사인과 코사인을 포함하는 함수, 사인/코사인과 탄젠트/코탄젠트를 포함하는 함수들의 적분 공식을 포함한다. 또한 대칭성을 이용한 정적분과 특정 구간에서의 정적분 계산을 위한 공식도 제시한다.

2. 사인 함수의 적분

일반적인 사인 함수의 적분 공식은 다음과 같다.

:$\backslash int\backslash sin\; ax\backslash ,dx\; =\; -\backslash frac\{1\}\{a\}\backslash cos\; ax+C$
:$\backslash int\backslash sin^2\; \{ax\}\backslash ,dx\; =\; \backslash frac\{x\}\{2\}\; -\; \backslash frac\{1\}\{4a\}\; \backslash sin\; 2ax\; +C=\; \backslash frac\{x\}\{2\}\; -\; \backslash frac\{1\}\{2a\}\; \backslash sin\; ax\backslash cos\; ax\; +C$
:$\backslash int\backslash sin^3\; \{ax\}\backslash ,dx\; =\; \backslash frac\{\backslash cos\; 3ax\}\{12a\}\; -\; \backslash frac\{3\; \backslash cos\; ax\}\{4a\}\; +C$
:$\backslash int\; x\backslash sin^2\; \{ax\}\backslash ,dx\; =\; \backslash frac\{x^2\}\{4\}\; -\; \backslash frac\{x\}\{4a\}\; \backslash sin\; 2ax\; -\; \backslash frac\{1\}\{8a^2\}\; \backslash cos\; 2ax\; +C$
:$\backslash int\; x^2\backslash sin^2\; \{ax\}\backslash ,dx\; =\; \backslash frac\{x^3\}\{6\}\; -\; \backslash left(\; \backslash frac\; \{x^2\}\{4a\}\; -\; \backslash frac\{1\}\{8a^3\}\; \backslash right)\; \backslash sin\; 2ax\; -\; \backslash frac\{x\}\{4a^2\}\; \backslash cos\; 2ax\; +C$
:$\backslash int\; x\backslash sin\; ax\backslash ,dx\; =\; \backslash frac\{\backslash sin\; ax\}\{a^2\}-\backslash frac\{x\backslash cos\; ax\}\{a\}+C$
:$\backslash int(\backslash sin\; b\_1x)(\backslash sin\; b\_2x)\backslash ,dx\; =\; \backslash frac\{\backslash sin((b\_2-b\_1)x)\}\{2(b\_2-b\_1)\}-\backslash frac\{\backslash sin((b\_1+b\_2)x)\}\{2(b\_1+b\_2)\}+C\; \backslash qquad\backslash mbox\{(for\; \}|b\_1|\backslash neq|b\_2|\backslash mbox\{)\}$
:$\backslash int\backslash sin^n\; \{ax\}\backslash ,dx\; =\; -\backslash frac\{\backslash sin^\{n-1\}\; ax\backslash cos\; ax\}\{na\}\; +\; \backslash frac\{n-1\}\{n\}\backslash int\backslash sin^\{n-2\}\; ax\backslash ,dx\; \backslash qquad\backslash mbox\{(for\; \}n>0\backslash mbox\{)\}$
:$\backslash int\backslash frac\{dx\}\{\backslash sin\; ax\}\; =\; -\backslash frac\{1\}\{a\}\backslash ln\{\backslash left|\; \backslash csc\{ax\}+\backslash cot\{ax\}\backslash right|\}+C$
:$\backslash int\backslash frac\{dx\}\{\backslash sin^n\; ax\}\; =\; \backslash frac\{\backslash cos\; ax\}\{a(1-n)\; \backslash sin^\{n-1\}\; ax\}+\backslash frac\{n-2\}\{n-1\}\backslash int\backslash frac\{dx\}\{\backslash sin^\{n-2\}ax\}\; \backslash qquad\backslash mbox\{(for\; \}n>1\backslash mbox\{)\}$
:$\backslash begin\{align\}$
\int x^n\sin ax\,dx &= -\frac{x^n}{a}\cos ax+\frac{n}{a}\int x^{n-1}\cos ax\,dx \\
&= \sum_{k=0}^{2k\leq n} (-1)^{k+1} \frac{x^{n-2k}}{a^{1+2k}}\frac{n!}{(n-2k)!} \cos ax +\sum_{k=0}^{2k+1\leq n}(-1)^k \frac{x^{n-1-2k}}{a^{2+2k}}\frac{n!}{(n-2k-1)!} \sin ax \\
&= - \sum_{k=0}^n \frac{x^{n-k}}{a^{1+k}}\frac{n!}{(n-k)!}\cos\left(ax+k\frac{\pi}{2}\right) \qquad\mbox{(for }n>0\mbox{)}
\end{align}
:$\backslash int\backslash frac\{\backslash sin\; ax\}\{x\}\backslash ,dx\; =\; \backslash sum\_\{n=0\}^\backslash infty\; (-1)^n\backslash frac\{(ax)^\{2n+1\}\}\{(2n+1)\backslash cdot\; (2n+1)!\}\; +C$
:$\backslash int\backslash frac\{\backslash sin\; ax\}\{x^n\}\backslash ,dx\; =\; -\backslash frac\{\backslash sin\; ax\}\{(n-1)x^\{n-1\}\}\; +\; \backslash frac\{a\}\{n-1\}\backslash int\backslash frac\{\backslash cos\; ax\}\{x^\{n-1\}\}\backslash ,dx$
:$\backslash int\backslash frac\{dx\}\{1\backslash pm\backslash sin\; ax\}\; =\; \backslash frac\{1\}\{a\}\backslash tan\backslash left(\backslash frac\{ax\}\{2\}\backslash mp\backslash frac\{\backslash pi\}\{4\}\backslash right)+C$
:$\backslash int\backslash frac\{x\backslash ,dx\}\{1+\backslash sin\; ax\}\; =\; \backslash frac\{x\}\{a\}\backslash tan\backslash left(\backslash frac\{ax\}\{2\}\; -\; \backslash frac\{\backslash pi\}\{4\}\backslash right)+\backslash frac\{2\}\{a^2\}\backslash ln\backslash left|\backslash cos\backslash left(\backslash frac\{ax\}\{2\}-\backslash frac\{\backslash pi\}\{4\}\backslash right)\backslash right|+C$
:$\backslash int\backslash frac\{x\backslash ,dx\}\{1-\backslash sin\; ax\}\; =\; \backslash frac\{x\}\{a\}\backslash cot\backslash left(\backslash frac\{\backslash pi\}\{4\}\; -\; \backslash frac\{ax\}\{2\}\backslash right)+\backslash frac\{2\}\{a^2\}\backslash ln\backslash left|\backslash sin\backslash left(\backslash frac\{\backslash pi\}\{4\}-\backslash frac\{ax\}\{2\}\backslash right)\backslash right|+C$
:$\backslash int\backslash frac\{\backslash sin\; ax\backslash ,dx\}\{1\backslash pm\backslash sin\; ax\}\; =\; \backslash pm\; x+\backslash frac\{1\}\{a\}\backslash tan\backslash left(\backslash frac\{\backslash pi\}\{4\}\backslash mp\backslash frac\{ax\}\{2\}\backslash right)+C$

위의 적분 공식들은 삼각함수의 적분 계산에 유용하게 활용될 수 있다.

2.1. 일반적인 사인 함수의 적분

일반적인 사인 함수의 적분 공식은 다음과 같다.

:$\backslash int\backslash sin\; ax\backslash ,dx\; =\; -\backslash frac\{1\}\{a\}\backslash cos\; ax+C$
:$\backslash int\backslash sin^2\; \{ax\}\backslash ,dx\; =\; \backslash frac\{x\}\{2\}\; -\; \backslash frac\{1\}\{4a\}\; \backslash sin\; 2ax\; +C=\; \backslash frac\{x\}\{2\}\; -\; \backslash frac\{1\}\{2a\}\; \backslash sin\; ax\backslash cos\; ax\; +C$
:$\backslash int\backslash sin^3\; \{ax\}\backslash ,dx\; =\; \backslash frac\{\backslash cos\; 3ax\}\{12a\}\; -\; \backslash frac\{3\; \backslash cos\; ax\}\{4a\}\; +C$
:$\backslash int\; x\backslash sin^2\; \{ax\}\backslash ,dx\; =\; \backslash frac\{x^2\}\{4\}\; -\; \backslash frac\{x\}\{4a\}\; \backslash sin\; 2ax\; -\; \backslash frac\{1\}\{8a^2\}\; \backslash cos\; 2ax\; +C$
:$\backslash int\; x^2\backslash sin^2\; \{ax\}\backslash ,dx\; =\; \backslash frac\{x^3\}\{6\}\; -\; \backslash left(\; \backslash frac\; \{x^2\}\{4a\}\; -\; \backslash frac\{1\}\{8a^3\}\; \backslash right)\; \backslash sin\; 2ax\; -\; \backslash frac\{x\}\{4a^2\}\; \backslash cos\; 2ax\; +C$
:$\backslash int\; x\backslash sin\; ax\backslash ,dx\; =\; \backslash frac\{\backslash sin\; ax\}\{a^2\}-\backslash frac\{x\backslash cos\; ax\}\{a\}+C$
:$\backslash int(\backslash sin\; b\_1x)(\backslash sin\; b\_2x)\backslash ,dx\; =\; \backslash frac\{\backslash sin((b\_2-b\_1)x)\}\{2(b\_2-b\_1)\}-\backslash frac\{\backslash sin((b\_1+b\_2)x)\}\{2(b\_1+b\_2)\}+C\; \backslash qquad\backslash mbox\{(for\; \}|b\_1|\backslash neq|b\_2|\backslash mbox\{)\}$
:$\backslash int\backslash sin^n\; \{ax\}\backslash ,dx\; =\; -\backslash frac\{\backslash sin^\{n-1\}\; ax\backslash cos\; ax\}\{na\}\; +\; \backslash frac\{n-1\}\{n\}\backslash int\backslash sin^\{n-2\}\; ax\backslash ,dx\; \backslash qquad\backslash mbox\{(for\; \}n>0\backslash mbox\{)\}$
:$\backslash int\backslash frac\{dx\}\{\backslash sin\; ax\}\; =\; -\backslash frac\{1\}\{a\}\backslash ln\{\backslash left|\; \backslash csc\{ax\}+\backslash cot\{ax\}\backslash right|\}+C$
:$\backslash int\backslash frac\{dx\}\{\backslash sin^n\; ax\}\; =\; \backslash frac\{\backslash cos\; ax\}\{a(1-n)\; \backslash sin^\{n-1\}\; ax\}+\backslash frac\{n-2\}\{n-1\}\backslash int\backslash frac\{dx\}\{\backslash sin^\{n-2\}ax\}\; \backslash qquad\backslash mbox\{(for\; \}n>1\backslash mbox\{)\}$
:$\backslash begin\{align\}$
\int x^n\sin ax\,dx &= -\frac{x^n}{a}\cos ax+\frac{n}{a}\int x^{n-1}\cos ax\,dx \\
&= \sum_{k=0}^{2k\leq n} (-1)^{k+1} \frac{x^{n-2k}}{a^{1+2k}}\frac{n!}{(n-2k)!} \cos ax +\sum_{k=0}^{2k+1\leq n}(-1)^k \frac{x^{n-1-2k}}{a^{2+2k}}\frac{n!}{(n-2k-1)!} \sin ax \\
&= - \sum_{k=0}^n \frac{x^{n-k}}{a^{1+k}}\frac{n!}{(n-k)!}\cos\left(ax+k\frac{\pi}{2}\right) \qquad\mbox{(for }n>0\mbox{)}
\end{align}
:$\backslash int\backslash frac\{\backslash sin\; ax\}\{x\}\backslash ,dx\; =\; \backslash sum\_\{n=0\}^\backslash infty\; (-1)^n\backslash frac\{(ax)^\{2n+1\}\}\{(2n+1)\backslash cdot\; (2n+1)!\}\; +C$
:$\backslash int\backslash frac\{\backslash sin\; ax\}\{x^n\}\backslash ,dx\; =\; -\backslash frac\{\backslash sin\; ax\}\{(n-1)x^\{n-1\}\}\; +\; \backslash frac\{a\}\{n-1\}\backslash int\backslash frac\{\backslash cos\; ax\}\{x^\{n-1\}\}\backslash ,dx$
:$\backslash int\backslash frac\{dx\}\{1\backslash pm\backslash sin\; ax\}\; =\; \backslash frac\{1\}\{a\}\backslash tan\backslash left(\backslash frac\{ax\}\{2\}\backslash mp\backslash frac\{\backslash pi\}\{4\}\backslash right)+C$
:$\backslash int\backslash frac\{x\backslash ,dx\}\{1+\backslash sin\; ax\}\; =\; \backslash frac\{x\}\{a\}\backslash tan\backslash left(\backslash frac\{ax\}\{2\}\; -\; \backslash frac\{\backslash pi\}\{4\}\backslash right)+\backslash frac\{2\}\{a^2\}\backslash ln\backslash left|\backslash cos\backslash left(\backslash frac\{ax\}\{2\}-\backslash frac\{\backslash pi\}\{4\}\backslash right)\backslash right|+C$
:$\backslash int\backslash frac\{x\backslash ,dx\}\{1-\backslash sin\; ax\}\; =\; \backslash frac\{x\}\{a\}\backslash cot\backslash left(\backslash frac\{\backslash pi\}\{4\}\; -\; \backslash frac\{ax\}\{2\}\backslash right)+\backslash frac\{2\}\{a^2\}\backslash ln\backslash left|\backslash sin\backslash left(\backslash frac\{\backslash pi\}\{4\}-\backslash frac\{ax\}\{2\}\backslash right)\backslash right|+C$
:$\backslash int\backslash frac\{\backslash sin\; ax\backslash ,dx\}\{1\backslash pm\backslash sin\; ax\}\; =\; \backslash pm\; x+\backslash frac\{1\}\{a\}\backslash tan\backslash left(\backslash frac\{\backslash pi\}\{4\}\backslash mp\backslash frac\{ax\}\{2\}\backslash right)+C$

위의 적분 공식들은 삼각함수의 적분 계산에 유용하게 활용될 수 있다.

2.2. 거듭제곱 사인 함수의 적분

사인 함수의 거듭제곱 형태에 대한 적분 공식은 다음과 같다.

* $\backslash int\backslash sin\; ax\backslash ,dx\; =\; -\backslash frac\{1\}\{a\}\backslash cos\; ax+C$
* $\backslash int\backslash sin^2\; \{ax\}\backslash ,dx\; =\; \backslash frac\{x\}\{2\}\; -\; \backslash frac\{1\}\{4a\}\; \backslash sin\; 2ax\; +C=\; \backslash frac\{x\}\{2\}\; -\; \backslash frac\{1\}\{2a\}\; \backslash sin\; ax\backslash cos\; ax\; +C$
* $\backslash int\backslash sin^3\; \{ax\}\backslash ,dx\; =\; \backslash frac\{\backslash cos\; 3ax\}\{12a\}\; -\; \backslash frac\{3\; \backslash cos\; ax\}\{4a\}\; +C$
* $\backslash int\; x\backslash sin^2\; \{ax\}\backslash ,dx\; =\; \backslash frac\{x^2\}\{4\}\; -\; \backslash frac\{x\}\{4a\}\; \backslash sin\; 2ax\; -\; \backslash frac\{1\}\{8a^2\}\; \backslash cos\; 2ax\; +C$
* $\backslash int\; x^2\backslash sin^2\; \{ax\}\backslash ,dx\; =\; \backslash frac\{x^3\}\{6\}\; -\; \backslash left(\; \backslash frac\; \{x^2\}\{4a\}\; -\; \backslash frac\{1\}\{8a^3\}\; \backslash right)\; \backslash sin\; 2ax\; -\; \backslash frac\{x\}\{4a^2\}\; \backslash cos\; 2ax\; +C$
* $\backslash int\; x\backslash sin\; ax\backslash ,dx\; =\; \backslash frac\{\backslash sin\; ax\}\{a^2\}-\backslash frac\{x\backslash cos\; ax\}\{a\}+C$
* $\backslash int(\backslash sin\; b\_1x)(\backslash sin\; b\_2x)\backslash ,dx\; =\; \backslash frac\{\backslash sin((b\_2-b\_1)x)\}\{2(b\_2-b\_1)\}-\backslash frac\{\backslash sin((b\_1+b\_2)x)\}\{2(b\_1+b\_2)\}+C\; \backslash qquad\backslash mbox\{(for\; \}|b\_1|\backslash neq|b\_2|\backslash mbox\{)\}$
* $\backslash int\backslash sin^n\; \{ax\}\backslash ,dx\; =\; -\backslash frac\{\backslash sin^\{n-1\}\; ax\backslash cos\; ax\}\{na\}\; +\; \backslash frac\{n-1\}\{n\}\backslash int\backslash sin^\{n-2\}\; ax\backslash ,dx\; \backslash qquad\backslash mbox\{(for\; \}n>0\backslash mbox\{)\}$
* $\backslash int\backslash frac\{dx\}\{\backslash sin\; ax\}\; =\; -\backslash frac\{1\}\{a\}\backslash ln\{\backslash left|\; \backslash csc\{ax\}+\backslash cot\{ax\}\backslash right|\}+C$
* $\backslash int\backslash frac\{dx\}\{\backslash sin^n\; ax\}\; =\; \backslash frac\{\backslash cos\; ax\}\{a(1-n)\; \backslash sin^\{n-1\}\; ax\}+\backslash frac\{n-2\}\{n-1\}\backslash int\backslash frac\{dx\}\{\backslash sin^\{n-2\}ax\}\; \backslash qquad\backslash mbox\{(for\; \}n>1\backslash mbox\{)\}$
* $\backslash begin\{align\}$
\int x^n\sin ax\,dx &= -\frac{x^n}{a}\cos ax+\frac{n}{a}\int x^{n-1}\cos ax\,dx \\
&= \sum_{k=0}^{2k\leq n} (-1)^{k+1} \frac{x^{n-2k}}{a^{1+2k}}\frac{n!}{(n-2k)!} \cos ax +\sum_{k=0}^{2k+1\leq n}(-1)^k \frac{x^{n-1-2k}}{a^{2+2k}}\frac{n!}{(n-2k-1)!} \sin ax \\
&= - \sum_{k=0}^n \frac{x^{n-k}}{a^{1+k}}\frac{n!}{(n-k)!}\cos\left(ax+k\frac{\pi}{2}\right) \qquad\mbox{(for }n>0\mbox{)}
\end{align}
* $\backslash int\backslash frac\{\backslash sin\; ax\}\{x\}\backslash ,dx\; =\; \backslash sum\_\{n=0\}^\backslash infty\; (-1)^n\backslash frac\{(ax)^\{2n+1\}\}\{(2n+1)\backslash cdot\; (2n+1)!\}\; +C$
* $\backslash int\backslash frac\{\backslash sin\; ax\}\{x^n\}\backslash ,dx\; =\; -\backslash frac\{\backslash sin\; ax\}\{(n-1)x^\{n-1\}\}\; +\; \backslash frac\{a\}\{n-1\}\backslash int\backslash frac\{\backslash cos\; ax\}\{x^\{n-1\}\}\backslash ,dx$
* $\backslash int\backslash frac\{dx\}\{1\backslash pm\backslash sin\; ax\}\; =\; \backslash frac\{1\}\{a\}\backslash tan\backslash left(\backslash frac\{ax\}\{2\}\backslash mp\backslash frac\{\backslash pi\}\{4\}\backslash right)+C$
* $\backslash int\backslash frac\{x\backslash ,dx\}\{1+\backslash sin\; ax\}\; =\; \backslash frac\{x\}\{a\}\backslash tan\backslash left(\backslash frac\{ax\}\{2\}\; -\; \backslash frac\{\backslash pi\}\{4\}\backslash right)+\backslash frac\{2\}\{a^2\}\backslash ln\backslash left|\backslash cos\backslash left(\backslash frac\{ax\}\{2\}-\backslash frac\{\backslash pi\}\{4\}\backslash right)\backslash right|+C$
* $\backslash int\backslash frac\{x\backslash ,dx\}\{1-\backslash sin\; ax\}\; =\; \backslash frac\{x\}\{a\}\backslash cot\backslash left(\backslash frac\{\backslash pi\}\{4\}\; -\; \backslash frac\{ax\}\{2\}\backslash right)+\backslash frac\{2\}\{a^2\}\backslash ln\backslash left|\backslash sin\backslash left(\backslash frac\{\backslash pi\}\{4\}-\backslash frac\{ax\}\{2\}\backslash right)\backslash right|+C$
* $\backslash int\backslash frac\{\backslash sin\; ax\backslash ,dx\}\{1\backslash pm\backslash sin\; ax\}\; =\; \backslash pm\; x+\backslash frac\{1\}\{a\}\backslash tan\backslash left(\backslash frac\{\backslash pi\}\{4\}\backslash mp\backslash frac\{ax\}\{2\}\backslash right)+C$

2.3. x와 사인 함수의 곱의 적분

$\backslash int\; x\backslash sin\; ax\backslash ,dx$는 $\backslash frac\{\backslash sin\; ax\}\{a^2\}-\backslash frac\{x\backslash cos\; ax\}\{a\}+C$이다.

$\backslash int\; x^n\backslash sin\; ax\backslash ,dx$는 다음과 같이 나타낼 수 있다.

:$\backslash begin\{align\}$
\int x^n\sin ax\,dx &= -\frac{x^n}{a}\cos ax+\frac{n}{a}\int x^{n-1}\cos ax\,dx \\
&= \sum_{k=0}^{2k\leq n} (-1)^{k+1} \frac{x^{n-2k}}{a^{1+2k}}\frac{n!}{(n-2k)!} \cos ax +\sum_{k=0}^{2k+1\leq n}(-1)^k \frac{x^{n-1-2k}}{a^{2+2k}}\frac{n!}{(n-2k-1)!} \sin ax \\
&= - \sum_{k=0}^n \frac{x^{n-k}}{a^{1+k}}\frac{n!}{(n-k)!}\cos\left(ax+k\frac{\pi}{2}\right) \qquad\mbox{(for }n>0\mbox{)}
\end{align}

$\backslash int\; x\backslash sin^2\; \{ax\}\backslash ,dx\; =\; \backslash frac\{x^2\}\{4\}\; -\; \backslash frac\{x\}\{4a\}\; \backslash sin\; 2ax\; -\; \backslash frac\{1\}\{8a^2\}\; \backslash cos\; 2ax\; +C$이다.

$\backslash int\; x^2\backslash sin^2\; \{ax\}\backslash ,dx\; =\; \backslash frac\{x^3\}\{6\}\; -\; \backslash left(\; \backslash frac\; \{x^2\}\{4a\}\; -\; \backslash frac\{1\}\{8a^3\}\; \backslash right)\; \backslash sin\; 2ax\; -\; \backslash frac\{x\}\{4a^2\}\; \backslash cos\; 2ax\; +C$이다.

2.4. 여러 사인 함수의 곱의 적분

$\backslash int(\backslash sin\; b\_1x)(\backslash sin\; b\_2x)\backslash ,dx\; =\; \backslash frac\{\backslash sin((b\_2-b\_1)x)\}\{2(b\_2-b\_1)\}-\backslash frac\{\backslash sin((b\_1+b\_2)x)\}\{2(b\_1+b\_2)\}+C\; \backslash qquad\backslash mbox\{(for\; \}|b\_1|\backslash neq|b\_2|\backslash mbox\{)\}$

이는 두 사인 함수의 곱을 적분하는 공식으로, 삼각함수의 덧셈정리를 이용하여 유도할 수 있다.

2.5. 분수 형태의 사인 함수의 적분

분수 형태의 사인 함수 적분은 다음과 같다.

*$\backslash int\backslash frac\{dx\}\{\backslash sin\; ax\}\; =\; -\backslash frac\{1\}\{a\}\backslash ln\{\backslash left|\; \backslash csc\{ax\}+\backslash cot\{ax\}\backslash right|\}+C$
*$\backslash int\backslash frac\{dx\}\{\backslash sin^n\; ax\}\; =\; \backslash frac\{\backslash cos\; ax\}\{a(1-n)\; \backslash sin^\{n-1\}\; ax\}+\backslash frac\{n-2\}\{n-1\}\backslash int\backslash frac\{dx\}\{\backslash sin^\{n-2\}ax\}\; \backslash qquad\backslash mbox\{(for\; \}n>1\backslash mbox\{)\}$
*$\backslash int\backslash frac\{\backslash sin\; ax\}\{x\}\backslash ,dx\; =\; \backslash sum\_\{n=0\}^\backslash infty\; (-1)^n\backslash frac\{(ax)^\{2n+1\}\}\{(2n+1)\backslash cdot\; (2n+1)!\}\; +C$
*$\backslash int\backslash frac\{\backslash sin\; ax\}\{x^n\}\backslash ,dx\; =\; -\backslash frac\{\backslash sin\; ax\}\{(n-1)x^\{n-1\}\}\; +\; \backslash frac\{a\}\{n-1\}\backslash int\backslash frac\{\backslash cos\; ax\}\{x^\{n-1\}\}\backslash ,dx$
*$\backslash int\backslash frac\{dx\}\{1\backslash pm\backslash sin\; ax\}\; =\; \backslash frac\{1\}\{a\}\backslash tan\backslash left(\backslash frac\{ax\}\{2\}\backslash mp\backslash frac\{\backslash pi\}\{4\}\backslash right)+C$
*$\backslash int\backslash frac\{x\backslash ,dx\}\{1+\backslash sin\; ax\}\; =\; \backslash frac\{x\}\{a\}\backslash tan\backslash left(\backslash frac\{ax\}\{2\}\; -\; \backslash frac\{\backslash pi\}\{4\}\backslash right)+\backslash frac\{2\}\{a^2\}\backslash ln\backslash left|\backslash cos\backslash left(\backslash frac\{ax\}\{2\}-\backslash frac\{\backslash pi\}\{4\}\backslash right)\backslash right|+C$
*$\backslash int\backslash frac\{x\backslash ,dx\}\{1-\backslash sin\; ax\}\; =\; \backslash frac\{x\}\{a\}\backslash cot\backslash left(\backslash frac\{\backslash pi\}\{4\}\; -\; \backslash frac\{ax\}\{2\}\backslash right)+\backslash frac\{2\}\{a^2\}\backslash ln\backslash left|\backslash sin\backslash left(\backslash frac\{\backslash pi\}\{4\}-\backslash frac\{ax\}\{2\}\backslash right)\backslash right|+C$
*$\backslash int\backslash frac\{\backslash sin\; ax\backslash ,dx\}\{1\backslash pm\backslash sin\; ax\}\; =\; \backslash pm\; x+\backslash frac\{1\}\{a\}\backslash tan\backslash left(\backslash frac\{\backslash pi\}\{4\}\backslash mp\backslash frac\{ax\}\{2\}\backslash right)+C$

위의 적분 공식들은 분모 또는 분자에 사인 함수를 포함하는 분수 형태의 함수에 대한 적분 결과를 나타낸다. 예를 들어, $\backslash int\backslash frac\{dx\}\{\backslash sin\; ax\}$는 사인 함수의 역수의 적분을 나타내며, 그 결과는 $-\backslash frac\{1\}\{a\}\backslash ln\{\backslash left|\; \backslash csc\{ax\}+\backslash cot\{ax\}\backslash right|\}+C$이다. 다른 공식들도 마찬가지로 분수 형태의 사인 함수와 관련된 다양한 적분 결과를 보여준다.

3. 코사인 함수의 적분

일반적인 코사인 함수의 적분 공식은 다음과 같다.

* $\backslash int\backslash cos\; ax\backslash ,dx\; =\; \backslash frac\{1\}\{a\}\backslash sin\; ax+C$
* $\backslash int\backslash cos^2\; \{ax\}\backslash ,dx\; =\; \backslash frac\{x\}\{2\}\; +\; \backslash frac\{1\}\{4a\}\; \backslash sin\; 2ax\; +C\; =\; \backslash frac\{x\}\{2\}\; +\; \backslash frac\{1\}\{2a\}\; \backslash sin\; ax\backslash cos\; ax\; +C$
* $\backslash int\backslash cos^n\; ax\backslash ,dx\; =\; \backslash frac\{\backslash cos^\{n-1\}\; ax\backslash sin\; ax\}\{na\}\; +\; \backslash frac\{n-1\}\{n\}\backslash int\backslash cos^\{n-2\}\; ax\backslash ,dx\; \backslash qquad\backslash mbox\{(for\; \}n>0\backslash mbox\{)\}$
* $\backslash int\; x\backslash cos\; ax\backslash ,dx\; =\; \backslash frac\{\backslash cos\; ax\}\{a^2\}\; +\; \backslash frac\{x\backslash sin\; ax\}\{a\}+C$
* $\backslash int\; x^2\backslash cos^2\; \{ax\}\backslash ,dx\; =\; \backslash frac\{x^3\}\{6\}\; +\; \backslash left(\; \backslash frac\; \{x^2\}\{4a\}\; -\; \backslash frac\{1\}\{8a^3\}\; \backslash right)\; \backslash sin\; 2ax\; +\; \backslash frac\{x\}\{4a^2\}\; \backslash cos\; 2ax\; +C$
* $\backslash begin\{align\}$
\int x^n\cos ax\,dx &= \frac{x^n\sin ax}{a} - \frac{n}{a}\int x^{n-1}\sin ax\,dx \\
&= \sum_{k=0}^{2k+1\leq n} (-1)^{k} \frac{x^{n-2k-1}}{a^{2+2k}}\frac{n!}{(n-2k-1)!} \cos ax +\sum_{k=0}^{2k\leq n}(-1)^{k} \frac{x^{n-2k}}{a^{1+2k}}\frac{n!}{(n-2k)!} \sin ax \\
&=\sum_{k=0}^n (-1)^{\lfloor k/2 \rfloor} \frac{x^{n-k}}{a^{1+k}}\frac{n!}{(n-k)!}\cos\left(ax -\frac{(-1)^k+1}{2}\frac{\pi}{2}\right) \\
&=\sum_{k=0}^n \frac{x^{n-k}}{a^{1+k}}\frac{n!}{(n-k)!}\sin\left(ax+k\frac{\pi}{2}\right) \qquad\mbox{(for }n>0\mbox{)}
\end{align}
* $\backslash int\backslash frac\{\backslash cos\; ax\}\{x\}\backslash ,dx\; =\; \backslash ln|ax|+\backslash sum\_\{k=1\}^\backslash infty\; (-1)^k\backslash frac\{(ax)^\{2k\}\}\{2k\backslash cdot(2k)!\}+C$
* $\backslash int\backslash frac\{\backslash cos\; ax\}\{x^n\}\backslash ,dx\; =\; -\backslash frac\{\backslash cos\; ax\}\{(n-1)x^\{n-1\}\}-\backslash frac\{a\}\{n-1\}\backslash int\backslash frac\{\backslash sin\; ax\}\{x^\{n-1\}\}\backslash ,dx\; \backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$
* $\backslash int\backslash frac\{dx\}\{\backslash cos\; ax\}\; =\; \backslash frac\{1\}\{a\}\backslash ln\backslash left|\backslash tan\backslash left(\backslash frac\{ax\}\{2\}+\backslash frac\{\backslash pi\}\{4\}\backslash right)\backslash right|+C$
* $\backslash int\backslash frac\{dx\}\{\backslash cos^n\; ax\}\; =\; \backslash frac\{\backslash sin\; ax\}\{a(n-1)\; \backslash cos^\{n-1\}\; ax\}\; +\; \backslash frac\{n-2\}\{n-1\}\backslash int\backslash frac\{dx\}\{\backslash cos^\{n-2\}\; ax\}\; \backslash qquad\backslash mbox\{(for\; \}n>1\backslash mbox\{)\}$
* $\backslash int\backslash frac\{dx\}\{1+\backslash cos\; ax\}\; =\; \backslash frac\{1\}\{a\}\backslash tan\backslash frac\{ax\}\{2\}+C$
* $\backslash int\backslash frac\{dx\}\{1-\backslash cos\; ax\}\; =\; -\backslash frac\{1\}\{a\}\backslash cot\backslash frac\{ax\}\{2\}+C$
* $\backslash int\backslash frac\{x\backslash ,dx\}\{1+\backslash cos\; ax\}\; =\; \backslash frac\{x\}\{a\}\backslash tan\backslash frac\{ax\}\{2\}\; +\; \backslash frac\{2\}\{a^2\}\backslash ln\backslash left|\backslash cos\backslash frac\{ax\}\{2\}\backslash right|+C$
* $\backslash int\backslash frac\{x\backslash ,dx\}\{1-\backslash cos\; ax\}\; =\; -\backslash frac\{x\}\{a\}\backslash cot\backslash frac\{ax\}\{2\}+\backslash frac\{2\}\{a^2\}\backslash ln\backslash left|\backslash sin\backslash frac\{ax\}\{2\}\backslash right|+C$
* $\backslash int\backslash frac\{\backslash cos\; ax\backslash ,dx\}\{1+\backslash cos\; ax\}\; =\; x\; -\; \backslash frac\{1\}\{a\}\backslash tan\backslash frac\{ax\}\{2\}+C$
* $\backslash int\backslash frac\{\backslash cos\; ax\backslash ,dx\}\{1-\backslash cos\; ax\}\; =\; -x-\backslash frac\{1\}\{a\}\backslash cot\backslash frac\{ax\}\{2\}+C$
* $\backslash int(\backslash cos\; a\_1x)(\backslash cos\; a\_2x)\backslash ,dx\; =\; \backslash frac\{\backslash sin((a\_2-a\_1)x)\}\{2(a\_2-a\_1)\}+\backslash frac\{\backslash sin((a\_2+a\_1)x)\}\{2(a\_2+a\_1)\}+C\; \backslash qquad\backslash mbox\{(for\; \}|a\_1|\backslash neq|a\_2|\backslash mbox\{)\}$

3.1. 일반적인 코사인 함수의 적분

일반적인 코사인 함수의 적분 공식은 다음과 같다.

* $\backslash int\backslash cos\; ax\backslash ,dx\; =\; \backslash frac\{1\}\{a\}\backslash sin\; ax+C$
* $\backslash int\backslash cos^2\; \{ax\}\backslash ,dx\; =\; \backslash frac\{x\}\{2\}\; +\; \backslash frac\{1\}\{4a\}\; \backslash sin\; 2ax\; +C\; =\; \backslash frac\{x\}\{2\}\; +\; \backslash frac\{1\}\{2a\}\; \backslash sin\; ax\backslash cos\; ax\; +C$
* $\backslash int\backslash cos^n\; ax\backslash ,dx\; =\; \backslash frac\{\backslash cos^\{n-1\}\; ax\backslash sin\; ax\}\{na\}\; +\; \backslash frac\{n-1\}\{n\}\backslash int\backslash cos^\{n-2\}\; ax\backslash ,dx\; \backslash qquad\backslash mbox\{(for\; \}n>0\backslash mbox\{)\}$
* $\backslash int\; x\backslash cos\; ax\backslash ,dx\; =\; \backslash frac\{\backslash cos\; ax\}\{a^2\}\; +\; \backslash frac\{x\backslash sin\; ax\}\{a\}+C$
* $\backslash int\; x^2\backslash cos^2\; \{ax\}\backslash ,dx\; =\; \backslash frac\{x^3\}\{6\}\; +\; \backslash left(\; \backslash frac\; \{x^2\}\{4a\}\; -\; \backslash frac\{1\}\{8a^3\}\; \backslash right)\; \backslash sin\; 2ax\; +\; \backslash frac\{x\}\{4a^2\}\; \backslash cos\; 2ax\; +C$
* $\backslash begin\{align\}$
\int x^n\cos ax\,dx &= \frac{x^n\sin ax}{a} - \frac{n}{a}\int x^{n-1}\sin ax\,dx \\
&= \sum_{k=0}^{2k+1\leq n} (-1)^{k} \frac{x^{n-2k-1}}{a^{2+2k}}\frac{n!}{(n-2k-1)!} \cos ax +\sum_{k=0}^{2k\leq n}(-1)^{k} \frac{x^{n-2k}}{a^{1+2k}}\frac{n!}{(n-2k)!} \sin ax \\
&=\sum_{k=0}^n (-1)^{\lfloor k/2 \rfloor} \frac{x^{n-k}}{a^{1+k}}\frac{n!}{(n-k)!}\cos\left(ax -\frac{(-1)^k+1}{2}\frac{\pi}{2}\right) \\
&=\sum_{k=0}^n \frac{x^{n-k}}{a^{1+k}}\frac{n!}{(n-k)!}\sin\left(ax+k\frac{\pi}{2}\right) \qquad\mbox{(for }n>0\mbox{)}
\end{align}
* $\backslash int\backslash frac\{\backslash cos\; ax\}\{x\}\backslash ,dx\; =\; \backslash ln|ax|+\backslash sum\_\{k=1\}^\backslash infty\; (-1)^k\backslash frac\{(ax)^\{2k\}\}\{2k\backslash cdot(2k)!\}+C$
* $\backslash int\backslash frac\{\backslash cos\; ax\}\{x^n\}\backslash ,dx\; =\; -\backslash frac\{\backslash cos\; ax\}\{(n-1)x^\{n-1\}\}-\backslash frac\{a\}\{n-1\}\backslash int\backslash frac\{\backslash sin\; ax\}\{x^\{n-1\}\}\backslash ,dx\; \backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$
* $\backslash int\backslash frac\{dx\}\{\backslash cos\; ax\}\; =\; \backslash frac\{1\}\{a\}\backslash ln\backslash left|\backslash tan\backslash left(\backslash frac\{ax\}\{2\}+\backslash frac\{\backslash pi\}\{4\}\backslash right)\backslash right|+C$
* $\backslash int\backslash frac\{dx\}\{\backslash cos^n\; ax\}\; =\; \backslash frac\{\backslash sin\; ax\}\{a(n-1)\; \backslash cos^\{n-1\}\; ax\}\; +\; \backslash frac\{n-2\}\{n-1\}\backslash int\backslash frac\{dx\}\{\backslash cos^\{n-2\}\; ax\}\; \backslash qquad\backslash mbox\{(for\; \}n>1\backslash mbox\{)\}$
* $\backslash int\backslash frac\{dx\}\{1+\backslash cos\; ax\}\; =\; \backslash frac\{1\}\{a\}\backslash tan\backslash frac\{ax\}\{2\}+C$
* $\backslash int\backslash frac\{dx\}\{1-\backslash cos\; ax\}\; =\; -\backslash frac\{1\}\{a\}\backslash cot\backslash frac\{ax\}\{2\}+C$
* $\backslash int\backslash frac\{x\backslash ,dx\}\{1+\backslash cos\; ax\}\; =\; \backslash frac\{x\}\{a\}\backslash tan\backslash frac\{ax\}\{2\}\; +\; \backslash frac\{2\}\{a^2\}\backslash ln\backslash left|\backslash cos\backslash frac\{ax\}\{2\}\backslash right|+C$
* $\backslash int\backslash frac\{x\backslash ,dx\}\{1-\backslash cos\; ax\}\; =\; -\backslash frac\{x\}\{a\}\backslash cot\backslash frac\{ax\}\{2\}+\backslash frac\{2\}\{a^2\}\backslash ln\backslash left|\backslash sin\backslash frac\{ax\}\{2\}\backslash right|+C$
* $\backslash int\backslash frac\{\backslash cos\; ax\backslash ,dx\}\{1+\backslash cos\; ax\}\; =\; x\; -\; \backslash frac\{1\}\{a\}\backslash tan\backslash frac\{ax\}\{2\}+C$
* $\backslash int\backslash frac\{\backslash cos\; ax\backslash ,dx\}\{1-\backslash cos\; ax\}\; =\; -x-\backslash frac\{1\}\{a\}\backslash cot\backslash frac\{ax\}\{2\}+C$
* $\backslash int(\backslash cos\; a\_1x)(\backslash cos\; a\_2x)\backslash ,dx\; =\; \backslash frac\{\backslash sin((a\_2-a\_1)x)\}\{2(a\_2-a\_1)\}+\backslash frac\{\backslash sin((a\_2+a\_1)x)\}\{2(a\_2+a\_1)\}+C\; \backslash qquad\backslash mbox\{(for\; \}|a\_1|\backslash neq|a\_2|\backslash mbox\{)\}$

3.2. 거듭제곱 코사인 함수의 적분

코사인 함수의 거듭제곱 형태의 적분 공식은 다음과 같다.

*$\backslash int\backslash cos\; ax\backslash ,dx\; =\; \backslash frac\{1\}\{a\}\backslash sin\; ax+C$

*$\backslash int\backslash cos^2\; \{ax\}\backslash ,dx\; =\; \backslash frac\{x\}\{2\}\; +\; \backslash frac\{1\}\{4a\}\; \backslash sin\; 2ax\; +C\; =\; \backslash frac\{x\}\{2\}\; +\; \backslash frac\{1\}\{2a\}\; \backslash sin\; ax\backslash cos\; ax\; +C$

*$\backslash int\backslash cos^n\; ax\backslash ,dx\; =\; \backslash frac\{\backslash cos^\{n-1\}\; ax\backslash sin\; ax\}\{na\}\; +\; \backslash frac\{n-1\}\{n\}\backslash int\backslash cos^\{n-2\}\; ax\backslash ,dx\; \backslash qquad\backslash mbox\{(for\; \}n>0\backslash mbox\{)\}$

*$\backslash int\; x\backslash cos\; ax\backslash ,dx\; =\; \backslash frac\{\backslash cos\; ax\}\{a^2\}\; +\; \backslash frac\{x\backslash sin\; ax\}\{a\}+C$

*$\backslash int\; x^2\backslash cos^2\; \{ax\}\backslash ,dx\; =\; \backslash frac\{x^3\}\{6\}\; +\; \backslash left(\; \backslash frac\; \{x^2\}\{4a\}\; -\; \backslash frac\{1\}\{8a^3\}\; \backslash right)\; \backslash sin\; 2ax\; +\; \backslash frac\{x\}\{4a^2\}\; \backslash cos\; 2ax\; +C$

*$\backslash begin\{align\}$
\int x^n\cos ax\,dx &= \frac{x^n\sin ax}{a} - \frac{n}{a}\int x^{n-1}\sin ax\,dx \\
&= \sum_{k=0}^{2k+1\leq n} (-1)^{k} \frac{x^{n-2k-1}}{a^{2+2k}}\frac{n!}{(n-2k-1)!} \cos ax +\sum_{k=0}^{2k\leq n}(-1)^{k} \frac{x^{n-2k}}{a^{1+2k}}\frac{n!}{(n-2k)!} \sin ax \\
&=\sum_{k=0}^n (-1)^{\lfloor k/2 \rfloor} \frac{x^{n-k}}{a^{1+k}}\frac{n!}{(n-k)!}\cos\left(ax -\frac{(-1)^k+1}{2}\frac{\pi}{2}\right) \\
&=\sum_{k=0}^n \frac{x^{n-k}}{a^{1+k}}\frac{n!}{(n-k)!}\sin\left(ax+k\frac{\pi}{2}\right) \qquad\mbox{(for }n>0\mbox{)}
\end{align}

*$\backslash int\backslash frac\{\backslash cos\; ax\}\{x\}\backslash ,dx\; =\; \backslash ln|ax|+\backslash sum\_\{k=1\}^\backslash infty\; (-1)^k\backslash frac\{(ax)^\{2k\}\}\{2k\backslash cdot(2k)!\}+C$

*$\backslash int\backslash frac\{\backslash cos\; ax\}\{x^n\}\backslash ,dx\; =\; -\backslash frac\{\backslash cos\; ax\}\{(n-1)x^\{n-1\}\}-\backslash frac\{a\}\{n-1\}\backslash int\backslash frac\{\backslash sin\; ax\}\{x^\{n-1\}\}\backslash ,dx\; \backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$

*$\backslash int\backslash frac\{dx\}\{\backslash cos\; ax\}\; =\; \backslash frac\{1\}\{a\}\backslash ln\backslash left|\backslash tan\backslash left(\backslash frac\{ax\}\{2\}+\backslash frac\{\backslash pi\}\{4\}\backslash right)\backslash right|+C$

*$\backslash int\backslash frac\{dx\}\{\backslash cos^n\; ax\}\; =\; \backslash frac\{\backslash sin\; ax\}\{a(n-1)\; \backslash cos^\{n-1\}\; ax\}\; +\; \backslash frac\{n-2\}\{n-1\}\backslash int\backslash frac\{dx\}\{\backslash cos^\{n-2\}\; ax\}\; \backslash qquad\backslash mbox\{(for\; \}n>1\backslash mbox\{)\}$

*$\backslash int\backslash frac\{dx\}\{1+\backslash cos\; ax\}\; =\; \backslash frac\{1\}\{a\}\backslash tan\backslash frac\{ax\}\{2\}+C$

*$\backslash int\backslash frac\{dx\}\{1-\backslash cos\; ax\}\; =\; -\backslash frac\{1\}\{a\}\backslash cot\backslash frac\{ax\}\{2\}+C$

*$\backslash int\backslash frac\{x\backslash ,dx\}\{1+\backslash cos\; ax\}\; =\; \backslash frac\{x\}\{a\}\backslash tan\backslash frac\{ax\}\{2\}\; +\; \backslash frac\{2\}\{a^2\}\backslash ln\backslash left|\backslash cos\backslash frac\{ax\}\{2\}\backslash right|+C$

*$\backslash int\backslash frac\{x\backslash ,dx\}\{1-\backslash cos\; ax\}\; =\; -\backslash frac\{x\}\{a\}\backslash cot\backslash frac\{ax\}\{2\}+\backslash frac\{2\}\{a^2\}\backslash ln\backslash left|\backslash sin\backslash frac\{ax\}\{2\}\backslash right|+C$

*$\backslash int\backslash frac\{\backslash cos\; ax\backslash ,dx\}\{1+\backslash cos\; ax\}\; =\; x\; -\; \backslash frac\{1\}\{a\}\backslash tan\backslash frac\{ax\}\{2\}+C$

*$\backslash int\backslash frac\{\backslash cos\; ax\backslash ,dx\}\{1-\backslash cos\; ax\}\; =\; -x-\backslash frac\{1\}\{a\}\backslash cot\backslash frac\{ax\}\{2\}+C$

*$\backslash int(\backslash cos\; a\_1x)(\backslash cos\; a\_2x)\backslash ,dx\; =\; \backslash frac\{\backslash sin((a\_2-a\_1)x)\}\{2(a\_2-a\_1)\}+\backslash frac\{\backslash sin((a\_2+a\_1)x)\}\{2(a\_2+a\_1)\}+C\; \backslash qquad\backslash mbox\{(for\; \}|a\_1|\backslash neq|a\_2|\backslash mbox\{)\}$

3.3. x와 코사인 함수의 곱의 적분

$\backslash int\; x\backslash cos\; ax\backslash ,dx\; =\; \backslash frac\{\backslash cos\; ax\}\{a^2\}\; +\; \backslash frac\{x\backslash sin\; ax\}\{a\}+C$

$\backslash int\; x^2\backslash cos^2\; \{ax\}\backslash ,dx\; =\; \backslash frac\{x^3\}\{6\}\; +\; \backslash left(\; \backslash frac\; \{x^2\}\{4a\}\; -\; \backslash frac\{1\}\{8a^3\}\; \backslash right)\; \backslash sin\; 2ax\; +\; \backslash frac\{x\}\{4a^2\}\; \backslash cos\; 2ax\; +C$

$\backslash begin\{align\}$
\int x^n\cos ax\,dx &= \frac{x^n\sin ax}{a} - \frac{n}{a}\int x^{n-1}\sin ax\,dx \\
&= \sum_{k=0}^{2k+1\leq n} (-1)^{k} \frac{x^{n-2k-1}}{a^{2+2k}}\frac{n!}{(n-2k-1)!} \cos ax +\sum_{k=0}^{2k\leq n}(-1)^{k} \frac{x^{n-2k}}{a^{1+2k}}\frac{n!}{(n-2k)!} \sin ax \\
&=\sum_{k=0}^n (-1)^{\lfloor k/2 \rfloor} \frac{x^{n-k}}{a^{1+k}}\frac{n!}{(n-k)!}\cos\left(ax -\frac{(-1)^k+1}{2}\frac{\pi}{2}\right) \\
&=\sum_{k=0}^n \frac{x^{n-k}}{a^{1+k}}\frac{n!}{(n-k)!}\sin\left(ax+k\frac{\pi}{2}\right) \qquad\mbox{(for }n>0\mbox{)}
\end{align}

$\backslash int\backslash frac\{x\backslash ,dx\}\{1+\backslash cos\; ax\}\; =\; \backslash frac\{x\}\{a\}\backslash tan\backslash frac\{ax\}\{2\}\; +\; \backslash frac\{2\}\{a^2\}\backslash ln\backslash left|\backslash cos\backslash frac\{ax\}\{2\}\backslash right|+C$

$\backslash int\backslash frac\{x\backslash ,dx\}\{1-\backslash cos\; ax\}\; =\; -\backslash frac\{x\}\{a\}\backslash cot\backslash frac\{ax\}\{2\}+\backslash frac\{2\}\{a^2\}\backslash ln\backslash left|\backslash sin\backslash frac\{ax\}\{2\}\backslash right|+C$

3.4. 여러 코사인 함수의 곱의 적분

$|a\_1|\backslash neq|a\_2|$일 때, 여러 코사인 함수의 곱의 적분은 다음과 같이 구할 수 있다.

:$\backslash int(\backslash cos\; a\_1x)(\backslash cos\; a\_2x)\backslash ,dx\; =\; \backslash frac\{\backslash sin((a\_2-a\_1)x)\}\{2(a\_2-a\_1)\}+\backslash frac\{\backslash sin((a\_2+a\_1)x)\}\{2(a\_2+a\_1)\}+C$

3.5. 분수 형태의 코사인 함수의 적분

분수 형태의 코사인 함수 적분은 다음과 같다.

*$\backslash int\backslash frac\{\backslash cos\; ax\}\{x\}\backslash ,dx\; =\; \backslash ln|ax|+\backslash sum\_\{k=1\}^\backslash infty\; (-1)^k\backslash frac\{(ax)^\{2k\}\}\{2k\backslash cdot(2k)!\}+C$

*$\backslash int\backslash frac\{\backslash cos\; ax\}\{x^n\}\backslash ,dx\; =\; -\backslash frac\{\backslash cos\; ax\}\{(n-1)x^\{n-1\}\}-\backslash frac\{a\}\{n-1\}\backslash int\backslash frac\{\backslash sin\; ax\}\{x^\{n-1\}\}\backslash ,dx\; \backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$

*$\backslash int\backslash frac\{dx\}\{\backslash cos\; ax\}\; =\; \backslash frac\{1\}\{a\}\backslash ln\backslash left|\backslash tan\backslash left(\backslash frac\{ax\}\{2\}+\backslash frac\{\backslash pi\}\{4\}\backslash right)\backslash right|+C$

*$\backslash int\backslash frac\{dx\}\{\backslash cos^n\; ax\}\; =\; \backslash frac\{\backslash sin\; ax\}\{a(n-1)\; \backslash cos^\{n-1\}\; ax\}\; +\; \backslash frac\{n-2\}\{n-1\}\backslash int\backslash frac\{dx\}\{\backslash cos^\{n-2\}\; ax\}\; \backslash qquad\backslash mbox\{(for\; \}n>1\backslash mbox\{)\}$

*$\backslash int\backslash frac\{dx\}\{1+\backslash cos\; ax\}\; =\; \backslash frac\{1\}\{a\}\backslash tan\backslash frac\{ax\}\{2\}+C$

*$\backslash int\backslash frac\{dx\}\{1-\backslash cos\; ax\}\; =\; -\backslash frac\{1\}\{a\}\backslash cot\backslash frac\{ax\}\{2\}+C$

*$\backslash int\backslash frac\{x\backslash ,dx\}\{1+\backslash cos\; ax\}\; =\; \backslash frac\{x\}\{a\}\backslash tan\backslash frac\{ax\}\{2\}\; +\; \backslash frac\{2\}\{a^2\}\backslash ln\backslash left|\backslash cos\backslash frac\{ax\}\{2\}\backslash right|+C$

*$\backslash int\backslash frac\{x\backslash ,dx\}\{1-\backslash cos\; ax\}\; =\; -\backslash frac\{x\}\{a\}\backslash cot\backslash frac\{ax\}\{2\}+\backslash frac\{2\}\{a^2\}\backslash ln\backslash left|\backslash sin\backslash frac\{ax\}\{2\}\backslash right|+C$

*$\backslash int\backslash frac\{\backslash cos\; ax\backslash ,dx\}\{1+\backslash cos\; ax\}\; =\; x\; -\; \backslash frac\{1\}\{a\}\backslash tan\backslash frac\{ax\}\{2\}+C$

*$\backslash int\backslash frac\{\backslash cos\; ax\backslash ,dx\}\{1-\backslash cos\; ax\}\; =\; -x-\backslash frac\{1\}\{a\}\backslash cot\backslash frac\{ax\}\{2\}+C$

4. 탄젠트 함수의 적분

일반적인 탄젠트 함수의 적분 공식은 다음과 같다.

:$\backslash int\backslash tan\; ax\backslash ,dx\; =\; -\backslash frac\{1\}\{a\}\backslash ln|\backslash cos\; ax|+C\; =\; \backslash frac\{1\}\{a\}\backslash ln|\backslash sec\; ax|+C$

:$\backslash int\; \backslash tan^2\{x\}\; \backslash ,\; dx\; =\; \backslash tan\{x\}\; -\; x\; +C$

:$\backslash int\backslash tan^n\; ax\backslash ,dx\; =\; \backslash frac\{1\}\{a(n-1)\}\backslash tan^\{n-1\}\; ax-\backslash int\backslash tan^\{n-2\}\; ax\backslash ,dx\; \backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$

:$\backslash int\backslash frac\{dx\}\{q\; \backslash tan\; ax\; +\; p\}\; =\; \backslash frac\{1\}\{p^2\; +\; q^2\}(px\; +\; \backslash frac\{q\}\{a\}\backslash ln|q\backslash sin\; ax\; +\; p\backslash cos\; ax|)+C\; \backslash qquad\backslash mbox\{(for\; \}p^2\; +\; q^2\backslash neq\; 0\backslash mbox\{)\}$

:$\backslash int\backslash frac\{dx\}\{\backslash tan\; ax\; \backslash pm\; 1\}\; =\; \backslash pm\; \backslash frac\{x\}\{2\}\; +\; \backslash frac\{1\}\{2a\}\backslash ln|\backslash sin\; ax\; \backslash pm\; \backslash cos\; ax|+C$

:$\backslash int\backslash frac\{\backslash tan\; ax\backslash ,dx\}\{\backslash tan\; ax\; \backslash pm\; 1\}\; =\; \backslash frac\{x\}\{2\}\; \backslash mp\; \backslash frac\{1\}\{2a\}\backslash ln|\backslash sin\; ax\; \backslash pm\; \backslash cos\; ax|+C$

4.1. 일반적인 탄젠트 함수의 적분

일반적인 탄젠트 함수의 적분 공식은 다음과 같다.

:$\backslash int\backslash tan\; ax\backslash ,dx\; =\; -\backslash frac\{1\}\{a\}\backslash ln|\backslash cos\; ax|+C\; =\; \backslash frac\{1\}\{a\}\backslash ln|\backslash sec\; ax|+C$

:$\backslash int\; \backslash tan^2\{x\}\; \backslash ,\; dx\; =\; \backslash tan\{x\}\; -\; x\; +C$

:$\backslash int\backslash tan^n\; ax\backslash ,dx\; =\; \backslash frac\{1\}\{a(n-1)\}\backslash tan^\{n-1\}\; ax-\backslash int\backslash tan^\{n-2\}\; ax\backslash ,dx\; \backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$

:$\backslash int\backslash frac\{dx\}\{q\; \backslash tan\; ax\; +\; p\}\; =\; \backslash frac\{1\}\{p^2\; +\; q^2\}(px\; +\; \backslash frac\{q\}\{a\}\backslash ln|q\backslash sin\; ax\; +\; p\backslash cos\; ax|)+C\; \backslash qquad\backslash mbox\{(for\; \}p^2\; +\; q^2\backslash neq\; 0\backslash mbox\{)\}$

:$\backslash int\backslash frac\{dx\}\{\backslash tan\; ax\; \backslash pm\; 1\}\; =\; \backslash pm\; \backslash frac\{x\}\{2\}\; +\; \backslash frac\{1\}\{2a\}\backslash ln|\backslash sin\; ax\; \backslash pm\; \backslash cos\; ax|+C$

:$\backslash int\backslash frac\{\backslash tan\; ax\backslash ,dx\}\{\backslash tan\; ax\; \backslash pm\; 1\}\; =\; \backslash frac\{x\}\{2\}\; \backslash mp\; \backslash frac\{1\}\{2a\}\backslash ln|\backslash sin\; ax\; \backslash pm\; \backslash cos\; ax|+C$

4.2. 거듭제곱 탄젠트 함수의 적분

탄젠트 함수의 거듭제곱 형태의 적분 공식은 다음과 같다.

:$\backslash int\backslash tan\; ax\backslash ,dx\; =\; -\backslash frac\{1\}\{a\}\backslash ln|\backslash cos\; ax|+C\; =\; \backslash frac\{1\}\{a\}\backslash ln|\backslash sec\; ax|+C$

:$\backslash int\; \backslash tan^2\{x\}\; \backslash ,\; dx\; =\; \backslash tan\{x\}\; -\; x\; +C$

:$\backslash int\backslash tan^n\; ax\backslash ,dx\; =\; \backslash frac\{1\}\{a(n-1)\}\backslash tan^\{n-1\}\; ax-\backslash int\backslash tan^\{n-2\}\; ax\backslash ,dx\; \backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$

:$\backslash int\backslash frac\{dx\}\{q\; \backslash tan\; ax\; +\; p\}\; =\; \backslash frac\{1\}\{p^2\; +\; q^2\}(px\; +\; \backslash frac\{q\}\{a\}\backslash ln|q\backslash sin\; ax\; +\; p\backslash cos\; ax|)+C\; \backslash qquad\backslash mbox\{(for\; \}p^2\; +\; q^2\backslash neq\; 0\backslash mbox\{)\}$

:$\backslash int\backslash frac\{dx\}\{\backslash tan\; ax\; \backslash pm\; 1\}\; =\; \backslash pm\; \backslash frac\{x\}\{2\}\; +\; \backslash frac\{1\}\{2a\}\backslash ln|\backslash sin\; ax\; \backslash pm\; \backslash cos\; ax|+C$

:$\backslash int\backslash frac\{\backslash tan\; ax\backslash ,dx\}\{\backslash tan\; ax\; \backslash pm\; 1\}\; =\; \backslash frac\{x\}\{2\}\; \backslash mp\; \backslash frac\{1\}\{2a\}\backslash ln|\backslash sin\; ax\; \backslash pm\; \backslash cos\; ax|+C$

4.3. 분수 형태의 탄젠트 함수의 적분

분수 형태의 탄젠트 함수 적분은 다음과 같다.

: $\backslash int\backslash tan\; ax\backslash ,dx\; =\; -\backslash frac\{1\}\{a\}\backslash ln|\backslash cos\; ax|+C\; =\; \backslash frac\{1\}\{a\}\backslash ln|\backslash sec\; ax|+C$

: $\backslash int\; \backslash tan^2\{x\}\; \backslash ,\; dx\; =\; \backslash tan\{x\}\; -\; x\; +C$

: $\backslash int\backslash tan^n\; ax\backslash ,dx\; =\; \backslash frac\{1\}\{a(n-1)\}\backslash tan^\{n-1\}\; ax-\backslash int\backslash tan^\{n-2\}\; ax\backslash ,dx\; \backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$

: $\backslash int\backslash frac\{dx\}\{q\; \backslash tan\; ax\; +\; p\}\; =\; \backslash frac\{1\}\{p^2\; +\; q^2\}(px\; +\; \backslash frac\{q\}\{a\}\backslash ln|q\backslash sin\; ax\; +\; p\backslash cos\; ax|)+C\; \backslash qquad\backslash mbox\{(for\; \}p^2\; +\; q^2\backslash neq\; 0\backslash mbox\{)\}$

: $\backslash int\backslash frac\{dx\}\{\backslash tan\; ax\; \backslash pm\; 1\}\; =\; \backslash pm\; \backslash frac\{x\}\{2\}\; +\; \backslash frac\{1\}\{2a\}\backslash ln|\backslash sin\; ax\; \backslash pm\; \backslash cos\; ax|+C$

: $\backslash int\backslash frac\{\backslash tan\; ax\backslash ,dx\}\{\backslash tan\; ax\; \backslash pm\; 1\}\; =\; \backslash frac\{x\}\{2\}\; \backslash mp\; \backslash frac\{1\}\{2a\}\backslash ln|\backslash sin\; ax\; \backslash pm\; \backslash cos\; ax|+C$

5. 시컨트 함수의 적분

일반적인 시컨트 함수의 적분 공식은 다음과 같다.

:$\backslash int\; \backslash sec\{ax\}\; \backslash ,\; dx\; =\; \backslash frac\{1\}\{a\}\backslash ln\{\backslash left|\; \backslash sec\{ax\}\; +\; \backslash tan\{ax\}\backslash right|\}+C=\; \backslash frac\{1\}\{a\}\backslash ln\{\backslash left|\; \backslash tan\{\backslash left(\backslash frac\{ax\}\{2\}\; +\; \backslash frac\{\backslash pi\}\{4\}\; \backslash right)\}\backslash right|\}+C\; =\; \backslash frac\{1\}\{a\}\backslash operatorname\{artanh\}\{\backslash left(\backslash sin\{ax\}\backslash right)\}+C$

:$\backslash int\; \backslash sec^2\{x\}\; \backslash ,\; dx\; =\; \backslash tan\{x\}+C$

:$\backslash int\; \backslash sec^3\{x\}\; \backslash ,\; dx\; =\; \backslash frac\{1\}\{2\}\backslash sec\; x\; \backslash tan\; x\; +\; \backslash frac\{1\}\{2\}\backslash ln|\backslash sec\; x\; +\; \backslash tan\; x|\; +\; C.$

:$\backslash int\; \backslash sec^n\{ax\}\; \backslash ,\; dx\; =\; \backslash frac\{\backslash sec^\{n-2\}\{ax\}\; \backslash tan\; \{ax\}\}\{a(n-1)\}\; \backslash ,+\backslash ,\; \backslash frac\{n-2\}\{n-1\}\backslash int\; \backslash sec^\{n-2\}\{ax\}\; \backslash ,\; dx\; \backslash qquad\; \backslash mbox\{\; (for\; \}n\; \backslash ne\; 1\backslash mbox\{)\}$

:$\backslash int\; \backslash frac\{dx\}\{\backslash sec\{x\}\; +\; 1\}\; =\; x\; -\; \backslash tan\{\backslash frac\{x\}\{2\}\}+C$
:$\backslash int\; \backslash frac\{dx\}\{\backslash sec\{x\}\; -\; 1\}\; =\; -\; x\; -\; \backslash cot\{\backslash frac\{x\}\{2\}\}+C$
:$\backslash int\; \backslash frac\{\backslash sin\{x\}\; \}\{\backslash cos\{x\}\}\; =\; \backslash int\; \backslash tan\{x\}$
:$\backslash int\; (\backslash sec\; x)(\backslash tan\; x)\; \backslash ,\; dx\; =\; \backslash sec\; x\; +\; C$
:$\backslash int\; \backslash sec\{ax\}\; \backslash ,\; \backslash mathrm\{d\}x\; =\; \backslash frac\{1\}\{a\}\backslash ln\{\backslash left|\; \backslash sec\{ax\}\; +\; \backslash tan\{ax\}\backslash right|\}+C\; =\; \backslash frac\{1\}\{a\}\backslash operatorname\{gd\}^\{-1\}(ax)+C$ 여기서 $\backslash operatorname\{gd\}^\{-1\}x$는 구데르만 함수의 역함수이다.
:$\backslash int\; \backslash sec^n\{x\}\; \backslash ,\; \backslash mathrm\{d\}x\; =\; \backslash frac\{\backslash sec^\{n-2\}\{x\}\backslash tan\{x\}\}\{n-1\}\; \backslash ,+\backslash ,\; \backslash frac\{n-2\}\{n-1\}\backslash int\; \backslash sec^\{n-2\}\{x\}\backslash ,\backslash mathrm\{d\}x$

5.1. 일반적인 시컨트 함수의 적분

일반적인 시컨트 함수의 적분 공식은 다음과 같다.

:$\backslash int\; \backslash sec\{ax\}\; \backslash ,\; dx\; =\; \backslash frac\{1\}\{a\}\backslash ln\{\backslash left|\; \backslash sec\{ax\}\; +\; \backslash tan\{ax\}\backslash right|\}+C=\; \backslash frac\{1\}\{a\}\backslash ln\{\backslash left|\; \backslash tan\{\backslash left(\backslash frac\{ax\}\{2\}\; +\; \backslash frac\{\backslash pi\}\{4\}\; \backslash right)\}\backslash right|\}+C\; =\; \backslash frac\{1\}\{a\}\backslash operatorname\{artanh\}\{\backslash left(\backslash sin\{ax\}\backslash right)\}+C$

:$\backslash int\; \backslash sec^2\{x\}\; \backslash ,\; dx\; =\; \backslash tan\{x\}+C$

:$\backslash int\; \backslash sec^3\{x\}\; \backslash ,\; dx\; =\; \backslash frac\{1\}\{2\}\backslash sec\; x\; \backslash tan\; x\; +\; \backslash frac\{1\}\{2\}\backslash ln|\backslash sec\; x\; +\; \backslash tan\; x|\; +\; C.$

:$\backslash int\; \backslash sec^n\{ax\}\; \backslash ,\; dx\; =\; \backslash frac\{\backslash sec^\{n-2\}\{ax\}\; \backslash tan\; \{ax\}\}\{a(n-1)\}\; \backslash ,+\backslash ,\; \backslash frac\{n-2\}\{n-1\}\backslash int\; \backslash sec^\{n-2\}\{ax\}\; \backslash ,\; dx\; \backslash qquad\; \backslash mbox\{\; (for\; \}n\; \backslash ne\; 1\backslash mbox\{)\}$

:$\backslash int\; \backslash frac\{dx\}\{\backslash sec\{x\}\; +\; 1\}\; =\; x\; -\; \backslash tan\{\backslash frac\{x\}\{2\}\}+C$
:$\backslash int\; \backslash frac\{dx\}\{\backslash sec\{x\}\; -\; 1\}\; =\; -\; x\; -\; \backslash cot\{\backslash frac\{x\}\{2\}\}+C$
:$\backslash int\; \backslash frac\{\backslash sin\{x\}\; \}\{\backslash cos\{x\}\}\; =\; \backslash int\; \backslash tan\{x\}$
:$\backslash int\; (\backslash sec\; x)(\backslash tan\; x)\; \backslash ,\; dx\; =\; \backslash sec\; x\; +\; C$
:$\backslash int\; \backslash sec\{ax\}\; \backslash ,\; \backslash mathrm\{d\}x\; =\; \backslash frac\{1\}\{a\}\backslash ln\{\backslash left|\; \backslash sec\{ax\}\; +\; \backslash tan\{ax\}\backslash right|\}+C\; =\; \backslash frac\{1\}\{a\}\backslash operatorname\{gd\}^\{-1\}(ax)+C$
:$\backslash int\; \backslash sec^n\{x\}\; \backslash ,\; \backslash mathrm\{d\}x\; =\; \backslash frac\{\backslash sec^\{n-2\}\{x\}\backslash tan\{x\}\}\{n-1\}\; \backslash ,+\backslash ,\; \backslash frac\{n-2\}\{n-1\}\backslash int\; \backslash sec^\{n-2\}\{x\}\backslash ,\backslash mathrm\{d\}x$

5.2. 거듭제곱 시컨트 함수의 적분

$\backslash int\; \backslash sec\{ax\}\; \backslash ,\; dx\; =\; \backslash frac\{1\}\{a\}\backslash ln\{\backslash left|\; \backslash sec\{ax\}\; +\; \backslash tan\{ax\}\backslash right|\}+C=\; \backslash frac\{1\}\{a\}\backslash ln\{\backslash left|\; \backslash tan\{\backslash left(\backslash frac\{ax\}\{2\}\; +\; \backslash frac\{\backslash pi\}\{4\}\; \backslash right)\}\backslash right|\}+C\; =\; \backslash frac\{1\}\{a\}\backslash operatorname\{artanh\}\{\backslash left(\backslash sin\{ax\}\backslash right)\}+C$

$\backslash int\; \backslash sec^2\{x\}\; \backslash ,\; dx\; =\; \backslash tan\{x\}+C$

$\backslash int\; \backslash sec^3\{x\}\; \backslash ,\; dx\; =\; \backslash frac\{1\}\{2\}\backslash sec\; x\; \backslash tan\; x\; +\; \backslash frac\{1\}\{2\}\backslash ln|\backslash sec\; x\; +\; \backslash tan\; x|\; +\; C.$

$\backslash int\; \backslash sec^n\{ax\}\; \backslash ,\; dx\; =\; \backslash frac\{\backslash sec^\{n-2\}\{ax\}\; \backslash tan\; \{ax\}\}\{a(n-1)\}\; \backslash ,+\backslash ,\; \backslash frac\{n-2\}\{n-1\}\backslash int\; \backslash sec^\{n-2\}\{ax\}\; \backslash ,\; dx\; \backslash qquad\; \backslash mbox\{\; (for\; \}n\; \backslash ne\; 1\backslash mbox\{)\}$

$\backslash int\; \backslash sec^n\{x\}\; \backslash ,\; \backslash mathrm\{d\}x\; =\; \backslash frac\{\backslash sec^\{n-2\}\{x\}\backslash tan\{x\}\}\{n-1\}\; \backslash ,+\backslash ,\; \backslash frac\{n-2\}\{n-1\}\backslash int\; \backslash sec^\{n-2\}\{x\}\backslash ,\backslash mathrm\{d\}x$

$\backslash int\; \backslash frac\{dx\}\{\backslash sec\{x\}\; +\; 1\}\; =\; x\; -\; \backslash tan\{\backslash frac\{x\}\{2\}\}+C$

$\backslash int\; \backslash frac\{dx\}\{\backslash sec\{x\}\; -\; 1\}\; =\; -\; x\; -\; \backslash cot\{\backslash frac\{x\}\{2\}\}+C$

$\backslash int\; \backslash frac\{\backslash sin\{x\}\; \}\{\backslash cos\{x\}\}\; =\; \backslash int\; \backslash tan\{x\}$

$\backslash int\; (\backslash sec\; x)(\backslash tan\; x)\; \backslash ,\; dx\; =\; \backslash sec\; x\; +\; C$

5.3. 분수 형태의 시컨트 함수의 적분

$\backslash int\; \backslash sec\{ax\}\; \backslash ,\; dx\; =\; \backslash frac\{1\}\{a\}\backslash ln\{\backslash left|\; \backslash sec\{ax\}\; +\; \backslash tan\{ax\}\backslash right|\}+C=\; \backslash frac\{1\}\{a\}\backslash ln\{\backslash left|\; \backslash tan\{\backslash left(\backslash frac\{ax\}\{2\}\; +\; \backslash frac\{\backslash pi\}\{4\}\; \backslash right)\}\backslash right|\}+C\; =\; \backslash frac\{1\}\{a\}\backslash operatorname\{artanh\}\{\backslash left(\backslash sin\{ax\}\backslash right)\}+C$

$\backslash int\; \backslash sec^2\{x\}\; \backslash ,\; dx\; =\; \backslash tan\{x\}+C$

$\backslash int\; \backslash sec^3\{x\}\; \backslash ,\; dx\; =\; \backslash frac\{1\}\{2\}\backslash sec\; x\; \backslash tan\; x\; +\; \backslash frac\{1\}\{2\}\backslash ln|\backslash sec\; x\; +\; \backslash tan\; x|\; +\; C.$

$\backslash int\; \backslash sec^n\{ax\}\; \backslash ,\; dx\; =\; \backslash frac\{\backslash sec^\{n-2\}\{ax\}\; \backslash tan\; \{ax\}\}\{a(n-1)\}\; \backslash ,+\backslash ,\; \backslash frac\{n-2\}\{n-1\}\backslash int\; \backslash sec^\{n-2\}\{ax\}\; \backslash ,\; dx\; \backslash qquad\; \backslash mbox\{\; (for\; \}n\; \backslash ne\; 1\backslash mbox\{)\}$

$\backslash int\; \backslash sec^n\{x\}\; \backslash ,\; \backslash mathrm\{d\}x\; =\; \backslash frac\{\backslash sec^\{n-2\}\{x\}\backslash tan\{x\}\}\{n-1\}\; \backslash ,+\backslash ,\; \backslash frac\{n-2\}\{n-1\}\backslash int\; \backslash sec^\{n-2\}\{x\}\backslash ,\backslash mathrm\{d\}x$

$\backslash int\; \backslash frac\{dx\}\{\backslash sec\{x\}\; +\; 1\}\; =\; x\; -\; \backslash tan\{\backslash frac\{x\}\{2\}\}+C$
$\backslash int\; \backslash frac\{dx\}\{\backslash sec\{x\}\; -\; 1\}\; =\; -\; x\; -\; \backslash cot\{\backslash frac\{x\}\{2\}\}+C$

6. 코시컨트 함수의 적분

일반적인 코시컨트 함수의 적분 공식은 다음과 같다.

:$\backslash int\; \backslash csc\{ax\}\; \backslash ,\; dx=\; -\backslash frac\{1\}\{a\}\backslash ln\{\backslash left|\; \backslash csc\{ax\}+\backslash cot\{ax\}\backslash right|\}+C=\; \backslash frac\{1\}\{a\}\backslash ln\{\backslash left|\; \backslash csc\{ax\}-\backslash cot\{ax\}\backslash right|\}+C\; =\; \backslash frac\{1\}\{a\}\backslash ln\{\backslash left|\; \backslash tan\{\backslash left(\; \backslash frac\{ax\}\{2\}\; \backslash right)\}\backslash right|\}+C$

:$\backslash int\; \backslash csc^2\{x\}\; \backslash ,\; dx\; =\; -\backslash cot\{x\}+C$

:$\backslash int\; \backslash csc^3\{x\}\; \backslash ,\; dx\; =\; -\backslash frac\{1\}\{2\}\backslash csc\; x\; \backslash cot\; x\; -\; \backslash frac\{1\}\{2\}\backslash ln|\backslash csc\; x\; +\; \backslash cot\; x|\; +\; C\; =\; -\backslash frac\{1\}\{2\}\backslash csc\; x\; \backslash cot\; x\; +\; \backslash frac\{1\}\{2\}\backslash ln|\backslash csc\; x\; -\; \backslash cot\; x|\; +\; C$

:$\backslash int\; \backslash csc^n\{ax\}\; \backslash ,\; dx\; =\; -\backslash frac\{\backslash csc^\{n-2\}\{ax\}\; \backslash cot\{ax\}\}\{a(n-1)\}\; \backslash ,+\backslash ,\; \backslash frac\{n-2\}\{n-1\}\backslash int\; \backslash csc^\{n-2\}\{ax\}\; \backslash ,\; dx\; \backslash qquad\; \backslash mbox\{\; (for\; \}n\; \backslash ne\; 1\backslash mbox\{)\}$

:$\backslash int\; \backslash frac\{dx\}\{\backslash csc\{x\}\; +\; 1\}\; =\; x\; -\; \backslash frac\{2\}\{\backslash cot\{\backslash frac\{x\}\{2\}\}+1\}+C$

:$\backslash int\; \backslash frac\{dx\}\{\backslash csc\{x\}\; -\; 1\}\; =\; -\; x\; +\; \backslash frac\{2\}\{\backslash cot\{\backslash frac\{x\}\{2\}\}-1\}+C$

:$\backslash int(\backslash csc\; x)(\backslash cot\; x)\backslash ,dx=\; -\backslash csc\; x\; +\; C$

6.1. 일반적인 코시컨트 함수의 적분

일반적인 코시컨트 함수의 적분 공식은 다음과 같다.

:$\backslash int\; \backslash csc\{ax\}\; \backslash ,\; dx=\; -\backslash frac\{1\}\{a\}\backslash ln\{\backslash left|\; \backslash csc\{ax\}+\backslash cot\{ax\}\backslash right|\}+C=\; \backslash frac\{1\}\{a\}\backslash ln\{\backslash left|\; \backslash csc\{ax\}-\backslash cot\{ax\}\backslash right|\}+C\; =\; \backslash frac\{1\}\{a\}\backslash ln\{\backslash left|\; \backslash tan\{\backslash left(\; \backslash frac\{ax\}\{2\}\; \backslash right)\}\backslash right|\}+C$

:$\backslash int\; \backslash csc^2\{x\}\; \backslash ,\; dx\; =\; -\backslash cot\{x\}+C$

:$\backslash int\; \backslash csc^3\{x\}\; \backslash ,\; dx\; =\; -\backslash frac\{1\}\{2\}\backslash csc\; x\; \backslash cot\; x\; -\; \backslash frac\{1\}\{2\}\backslash ln|\backslash csc\; x\; +\; \backslash cot\; x|\; +\; C\; =\; -\backslash frac\{1\}\{2\}\backslash csc\; x\; \backslash cot\; x\; +\; \backslash frac\{1\}\{2\}\backslash ln|\backslash csc\; x\; -\; \backslash cot\; x|\; +\; C$

:$\backslash int\; \backslash csc^n\{ax\}\; \backslash ,\; dx\; =\; -\backslash frac\{\backslash csc^\{n-2\}\{ax\}\; \backslash cot\{ax\}\}\{a(n-1)\}\; \backslash ,+\backslash ,\; \backslash frac\{n-2\}\{n-1\}\backslash int\; \backslash csc^\{n-2\}\{ax\}\; \backslash ,\; dx\; \backslash qquad\; \backslash mbox\{\; (for\; \}n\; \backslash ne\; 1\backslash mbox\{)\}$

:$\backslash int\; \backslash frac\{dx\}\{\backslash csc\{x\}\; +\; 1\}\; =\; x\; -\; \backslash frac\{2\}\{\backslash cot\{\backslash frac\{x\}\{2\}\}+1\}+C$

:$\backslash int\; \backslash frac\{dx\}\{\backslash csc\{x\}\; -\; 1\}\; =\; -\; x\; +\; \backslash frac\{2\}\{\backslash cot\{\backslash frac\{x\}\{2\}\}-1\}+C$

:$\backslash int(\backslash csc\; x)(\backslash cot\; x)\backslash ,dx=\; -\backslash csc\; x\; +\; C$

6.2. 거듭제곱 코시컨트 함수의 적분

$\backslash int\; \backslash csc\{ax\}\; \backslash ,\; dx=\; -\backslash frac\{1\}\{a\}\backslash ln\{\backslash left|\; \backslash csc\{ax\}+\backslash cot\{ax\}\backslash right|\}+C=\; \backslash frac\{1\}\{a\}\backslash ln\{\backslash left|\; \backslash csc\{ax\}-\backslash cot\{ax\}\backslash right|\}+C\; =\; \backslash frac\{1\}\{a\}\backslash ln\{\backslash left|\; \backslash tan\{\backslash left(\; \backslash frac\{ax\}\{2\}\; \backslash right)\}\backslash right|\}+C$

$\backslash int\; \backslash csc^2\{x\}\; \backslash ,\; dx\; =\; -\backslash cot\{x\}+C$

$\backslash int\; \backslash csc^3\{x\}\; \backslash ,\; dx\; =\; -\backslash frac\{1\}\{2\}\backslash csc\; x\; \backslash cot\; x\; -\; \backslash frac\{1\}\{2\}\backslash ln|\backslash csc\; x\; +\; \backslash cot\; x|\; +\; C\; =\; -\backslash frac\{1\}\{2\}\backslash csc\; x\; \backslash cot\; x\; +\; \backslash frac\{1\}\{2\}\backslash ln|\backslash csc\; x\; -\; \backslash cot\; x|\; +\; C$

$\backslash int\; \backslash csc^n\{ax\}\; \backslash ,\; dx\; =\; -\backslash frac\{\backslash csc^\{n-2\}\{ax\}\; \backslash cot\{ax\}\}\{a(n-1)\}\; \backslash ,+\backslash ,\; \backslash frac\{n-2\}\{n-1\}\backslash int\; \backslash csc^\{n-2\}\{ax\}\; \backslash ,\; dx\; \backslash qquad\; \backslash mbox\{\; (for\; \}n\; \backslash ne\; 1\backslash mbox\{)\}$

$\backslash int\; \backslash frac\{dx\}\{\backslash csc\{x\}\; +\; 1\}\; =\; x\; -\; \backslash frac\{2\}\{\backslash cot\{\backslash frac\{x\}\{2\}\}+1\}+C$

$\backslash int\; \backslash frac\{dx\}\{\backslash csc\{x\}\; -\; 1\}\; =\; -\; x\; +\; \backslash frac\{2\}\{\backslash cot\{\backslash frac\{x\}\{2\}\}-1\}+C$

6.3. 분수 형태의 코시컨트 함수의 적분

$\backslash int\; \backslash csc\{ax\}\; \backslash ,\; dx=\; -\backslash frac\{1\}\{a\}\backslash ln\{\backslash left|\; \backslash csc\{ax\}+\backslash cot\{ax\}\backslash right|\}+C=\; \backslash frac\{1\}\{a\}\backslash ln\{\backslash left|\; \backslash csc\{ax\}-\backslash cot\{ax\}\backslash right|\}+C\; =\; \backslash frac\{1\}\{a\}\backslash ln\{\backslash left|\; \backslash tan\{\backslash left(\; \backslash frac\{ax\}\{2\}\; \backslash right)\}\backslash right|\}+C$

$\backslash int\; \backslash csc^2\{x\}\; \backslash ,\; dx\; =\; -\backslash cot\{x\}+C$

$\backslash int\; \backslash csc^3\{x\}\; \backslash ,\; dx\; =\; -\backslash frac\{1\}\{2\}\backslash csc\; x\; \backslash cot\; x\; -\; \backslash frac\{1\}\{2\}\backslash ln|\backslash csc\; x\; +\; \backslash cot\; x|\; +\; C\; =\; -\backslash frac\{1\}\{2\}\backslash csc\; x\; \backslash cot\; x\; +\; \backslash frac\{1\}\{2\}\backslash ln|\backslash csc\; x\; -\; \backslash cot\; x|\; +\; C$

$\backslash int\; \backslash csc^n\{ax\}\; \backslash ,\; dx\; =\; -\backslash frac\{\backslash csc^\{n-2\}\{ax\}\; \backslash cot\{ax\}\}\{a(n-1)\}\; \backslash ,+\backslash ,\; \backslash frac\{n-2\}\{n-1\}\backslash int\; \backslash csc^\{n-2\}\{ax\}\; \backslash ,\; dx\; \backslash qquad\; \backslash mbox\{\; (for\; \}n\; \backslash ne\; 1\backslash mbox\{)\}$

$\backslash int\; \backslash frac\{dx\}\{\backslash csc\{x\}\; +\; 1\}\; =\; x\; -\; \backslash frac\{2\}\{\backslash cot\{\backslash frac\{x\}\{2\}\}+1\}+C$

$\backslash int\; \backslash frac\{dx\}\{\backslash csc\{x\}\; -\; 1\}\; =\; -\; x\; +\; \backslash frac\{2\}\{\backslash cot\{\backslash frac\{x\}\{2\}\}-1\}+C$

7. 코탄젠트 함수의 적분

:$\backslash int\backslash cot\; ax\backslash ,dx\; =\; \backslash frac\{1\}\{a\}\backslash ln|\backslash sin\; ax|+C$

:$\backslash int\; \backslash cot^2\{x\}\; \backslash ,\; dx\; =\; -\backslash cot\{x\}\; -\; x\; +C$

:$\backslash int\backslash cot^n\; ax\backslash ,dx\; =\; -\backslash frac\{1\}\{a(n-1)\}\backslash cot^\{n-1\}\; ax\; -\; \backslash int\backslash cot^\{n-2\}\; ax\backslash ,dx\; \backslash qquad\backslash mbox\{(\}n\backslash neq\; 1\backslash mbox\{)\}\backslash ,\backslash !$

:$\backslash int\backslash frac\{dx\}\{1\; +\; \backslash cot\; ax\}\; =\; \backslash int\backslash frac\{\backslash tan\; ax\backslash ,dx\}\{\backslash tan\; ax+1\}\; =\; \backslash frac\{x\}\{2\}\; -\; \backslash frac\{1\}\{2a\}\backslash ln|\backslash sin\; ax\; +\; \backslash cos\; ax|+C$

:$\backslash int\backslash frac\{dx\}\{1\; -\; \backslash cot\; ax\}\; =\; \backslash int\backslash frac\{\backslash tan\; ax\backslash ,dx\}\{\backslash tan\; ax-1\}\; =\; \backslash frac\{x\}\{2\}\; +\; \backslash frac\{1\}\{2a\}\backslash ln|\backslash sin\; ax\; -\; \backslash cos\; ax|+C$

7.1. 일반적인 코탄젠트 함수의 적분

일반적인 코탄젠트 함수의 적분 공식은 다음과 같다.

*$\backslash int\backslash cot\; ax\backslash ,dx\; =\; \backslash frac\{1\}\{a\}\backslash ln|\backslash sin\; ax|+C$

*$\backslash int\; \backslash cot^2\{x\}\; \backslash ,\; dx\; =\; -\backslash cot\{x\}\; -\; x\; +C$

*$\backslash int\backslash cot^n\; ax\backslash ,dx\; =\; -\backslash frac\{1\}\{a(n-1)\}\backslash cot^\{n-1\}\; ax\; -\; \backslash int\backslash cot^\{n-2\}\; ax\backslash ,dx\; \backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$

*$\backslash int\backslash frac\{dx\}\{1\; +\; \backslash cot\; ax\}\; =\; \backslash int\backslash frac\{\backslash tan\; ax\backslash ,dx\}\{\backslash tan\; ax+1\}\; =\; \backslash frac\{x\}\{2\}\; -\; \backslash frac\{1\}\{2a\}\backslash ln|\backslash sin\; ax\; +\; \backslash cos\; ax|+C$

*$\backslash int\backslash frac\{dx\}\{1\; -\; \backslash cot\; ax\}\; =\; \backslash int\backslash frac\{\backslash tan\; ax\backslash ,dx\}\{\backslash tan\; ax-1\}\; =\; \backslash frac\{x\}\{2\}\; +\; \backslash frac\{1\}\{2a\}\backslash ln|\backslash sin\; ax\; -\; \backslash cos\; ax|+C$

7.2. 거듭제곱 코탄젠트 함수의 적분

$\backslash int\backslash cot\; ax\backslash ,dx\; =\; \backslash frac\{1\}\{a\}\backslash ln|\backslash sin\; ax|+C$

$\backslash int\; \backslash cot^2\{x\}\; \backslash ,\; dx\; =\; -\backslash cot\{x\}\; -\; x\; +C$

$\backslash int\backslash cot^n\; ax\backslash ,dx\; =\; -\backslash frac\{1\}\{a(n-1)\}\backslash cot^\{n-1\}\; ax\; -\; \backslash int\backslash cot^\{n-2\}\; ax\backslash ,dx\; \backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$

$\backslash int\backslash frac\{dx\}\{1\; +\; \backslash cot\; ax\}\; =\; \backslash int\backslash frac\{\backslash tan\; ax\backslash ,dx\}\{\backslash tan\; ax+1\}\; =\; \backslash frac\{x\}\{2\}\; -\; \backslash frac\{1\}\{2a\}\backslash ln|\backslash sin\; ax\; +\; \backslash cos\; ax|+C$

$\backslash int\backslash frac\{dx\}\{1\; -\; \backslash cot\; ax\}\; =\; \backslash int\backslash frac\{\backslash tan\; ax\backslash ,dx\}\{\backslash tan\; ax-1\}\; =\; \backslash frac\{x\}\{2\}\; +\; \backslash frac\{1\}\{2a\}\backslash ln|\backslash sin\; ax\; -\; \backslash cos\; ax|+C$

7.3. 분수 형태의 코탄젠트 함수의 적분

$\backslash int\backslash cot\; ax\backslash ,dx\; =\; \backslash frac\{1\}\{a\}\backslash ln|\backslash sin\; ax|+C$

$\backslash int\; \backslash cot^2\{x\}\; \backslash ,\; dx\; =\; -\backslash cot\{x\}\; -\; x\; +C$

$\backslash int\backslash cot^n\; ax\backslash ,dx\; =\; -\backslash frac\{1\}\{a(n-1)\}\backslash cot^\{n-1\}\; ax\; -\; \backslash int\backslash cot^\{n-2\}\; ax\backslash ,dx\; \backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$

$\backslash int\backslash frac\{dx\}\{1\; +\; \backslash cot\; ax\}\; =\; \backslash int\backslash frac\{\backslash tan\; ax\backslash ,dx\}\{\backslash tan\; ax+1\}\; =\; \backslash frac\{x\}\{2\}\; -\; \backslash frac\{1\}\{2a\}\backslash ln|\backslash sin\; ax\; +\; \backslash cos\; ax|+C$

$\backslash int\backslash frac\{dx\}\{1\; -\; \backslash cot\; ax\}\; =\; \backslash int\backslash frac\{\backslash tan\; ax\backslash ,dx\}\{\backslash tan\; ax-1\}\; =\; \backslash frac\{x\}\{2\}\; +\; \backslash frac\{1\}\{2a\}\backslash ln|\backslash sin\; ax\; -\; \backslash cos\; ax|+C$

8. 사인과 코사인을 포함하는 함수의 적분

사인과 코사인을 모두 포함하는 피적분 함수에 대한 적분 공식은 다음과 같다. Bioche의 규칙을 사용하여 사인과 코사인의 유리 함수인 적분을 평가할 수 있다.

* $\backslash int\backslash frac\{dx\}\{\backslash cos\; ax\backslash pm\backslash sin\; ax\}\; =\; \backslash frac\{1\}\{a\backslash sqrt\{2\}\}\backslash ln\backslash left|\backslash tan\backslash left(\backslash frac\{ax\}\{2\}\backslash pm\backslash frac\{\backslash pi\}\{8\}\backslash right)\backslash right|+C$
* $\backslash int\backslash frac\{dx\}\{(\backslash cos\; ax\backslash pm\backslash sin\; ax)^2\}\; =\; \backslash frac\{1\}\{2a\}\backslash tan\backslash left(ax\backslash mp\backslash frac\{\backslash pi\}\{4\}\backslash right)+C$
* $\backslash int\backslash frac\{dx\}\{(\backslash cos\; x\; +\; \backslash sin\; x)^n\}\; =\; \backslash frac\{1\}\{2(n-1)\}\backslash left(\backslash frac\{\backslash sin\; x\; -\; \backslash cos\; x\}\{(\backslash cos\; x\; +\; \backslash sin\; x)^\{n\; -\; 1\}\}\; +\; (n\; -\; 2)\backslash int\backslash frac\{dx\}\{(\backslash cos\; x\; +\; \backslash sin\; x)^\{n-2\}\}\; \backslash right)$
* $\backslash int\backslash frac\{\backslash cos\; ax\backslash ,dx\}\{\backslash cos\; ax\; +\; \backslash sin\; ax\}\; =\; \backslash frac\{x\}\{2\}\; +\; \backslash frac\{1\}\{2a\}\backslash ln\backslash left|\backslash sin\; ax\; +\; \backslash cos\; ax\backslash right|+C$
* $\backslash int\backslash frac\{\backslash cos\; ax\backslash ,dx\}\{\backslash cos\; ax\; -\; \backslash sin\; ax\}\; =\; \backslash frac\{x\}\{2\}\; -\; \backslash frac\{1\}\{2a\}\backslash ln\backslash left|\backslash sin\; ax\; -\; \backslash cos\; ax\backslash right|+C$
* $\backslash int\backslash frac\{\backslash sin\; ax\backslash ,dx\}\{\backslash cos\; ax\; +\; \backslash sin\; ax\}\; =\; \backslash frac\{x\}\{2\}\; -\; \backslash frac\{1\}\{2a\}\backslash ln\backslash left|\backslash sin\; ax\; +\; \backslash cos\; ax\backslash right|+C$
* $\backslash int\backslash frac\{\backslash sin\; ax\backslash ,dx\}\{\backslash cos\; ax\; -\; \backslash sin\; ax\}\; =\; -\backslash frac\{x\}\{2\}\; -\; \backslash frac\{1\}\{2a\}\backslash ln\backslash left|\backslash sin\; ax\; -\; \backslash cos\; ax\backslash right|+C$
* $\backslash int\backslash frac\{\backslash cos\; ax\backslash ,dx\}\{(\backslash sin\; ax)(1+\backslash cos\; ax)\}\; =\; -\backslash frac\{1\}\{4a\}\backslash tan^2\backslash frac\{ax\}\{2\}+\backslash frac\{1\}\{2a\}\backslash ln\backslash left|\backslash tan\backslash frac\{ax\}\{2\}\backslash right|+C$
* $\backslash int\backslash frac\{\backslash cos\; ax\backslash ,dx\}\{(\backslash sin\; ax)(1-\backslash cos\; ax)\}\; =\; -\backslash frac\{1\}\{4a\}\backslash cot^2\backslash frac\{ax\}\{2\}-\backslash frac\{1\}\{2a\}\backslash ln\backslash left|\backslash tan\backslash frac\{ax\}\{2\}\backslash right|+C$
* $\backslash int\backslash frac\{\backslash sin\; ax\backslash ,dx\}\{(\backslash cos\; ax)(1+\backslash sin\; ax)\}\; =\; \backslash frac\{1\}\{4a\}\backslash cot^2\backslash left(\backslash frac\{ax\}\{2\}+\backslash frac\{\backslash pi\}\{4\}\backslash right)+\backslash frac\{1\}\{2a\}\backslash ln\backslash left|\backslash tan\backslash left(\backslash frac\{ax\}\{2\}+\backslash frac\{\backslash pi\}\{4\}\backslash right)\backslash right|+C$
* $\backslash int\backslash frac\{\backslash sin\; ax\backslash ,dx\}\{(\backslash cos\; ax)(1-\backslash sin\; ax)\}\; =\; \backslash frac\{1\}\{4a\}\backslash tan^2\backslash left(\backslash frac\{ax\}\{2\}+\backslash frac\{\backslash pi\}\{4\}\backslash right)-\backslash frac\{1\}\{2a\}\backslash ln\backslash left|\backslash tan\backslash left(\backslash frac\{ax\}\{2\}+\backslash frac\{\backslash pi\}\{4\}\backslash right)\backslash right|+C$
* $\backslash int(\backslash sin\; ax)(\backslash cos\; ax)\backslash ,dx\; =\; \backslash frac\{1\}\{2a\}\backslash sin^2\; ax\; +C$
* $\backslash int(\backslash sin\; a\_1x)(\backslash cos\; a\_2x)\backslash ,dx\; =\; -\backslash frac\{\backslash cos((a\_1-a\_2)x)\}\{2(a\_1-a\_2)\}\; -\backslash frac\{\backslash cos((a\_1+a\_2)x)\}\{2(a\_1+a\_2)\}\; +C\backslash qquad\backslash mbox\{(for\; \}|a\_1|\backslash neq|a\_2|\backslash mbox\{)\}$
* $\backslash int(\backslash sin^n\; ax)(\backslash cos\; ax)\backslash ,dx\; =\; \backslash frac\{1\}\{a(n+1)\}\backslash sin^\{n+1\}\; ax\; +C\backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; -1\backslash mbox\{)\}$
* $\backslash int(\backslash sin\; ax)(\backslash cos^n\; ax)\backslash ,dx\; =\; -\backslash frac\{1\}\{a(n+1)\}\backslash cos^\{n+1\}\; ax\; +C\backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; -1\backslash mbox\{)\}$
* $\backslash begin\{cases\}$
\int(\sin^n ax)(\cos^m ax)\,dx = -\frac{(\sin^{n-1} ax)(\cos^{m+1} ax)}{a(n+m)}+\frac{n-1}{n+m}\int(\sin^{n-2} ax)(\cos^m ax)\,dx \qquad\mbox{(for }m,n>0\mbox{)} \\
\int(\sin^n ax)(\cos^m ax)\,dx = \frac{(\sin^{n+1} ax)(\cos^{m-1} ax)}{a(n+m)} + \frac{m-1}{n+m}\int(\sin^n ax)(\cos^{m-2} ax)\,dx \qquad\mbox{(for }m,n>0\mbox{)}
\end{cases}
* $\backslash int\backslash frac\{dx\}\{(\backslash sin\; ax)(\backslash cos\; ax)\}\; =\; \backslash frac\{1\}\{a\}\backslash ln\backslash left|\backslash tan\; ax\backslash right|+C$
* $\backslash int\backslash frac\{dx\}\{(\backslash sin\; ax)(\backslash cos^n\; ax)\}\; =\; \backslash frac\{1\}\{a(n-1)\backslash cos^\{n-1\}\; ax\}+\backslash int\backslash frac\{dx\}\{(\backslash sin\; ax)(\backslash cos^\{n-2\}\; ax)\}\; \backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$
* $\backslash int\backslash frac\{dx\}\{(\backslash sin^n\; ax)(\backslash cos\; ax)\}\; =\; -\backslash frac\{1\}\{a(n-1)\backslash sin^\{n-1\}\; ax\}+\backslash int\backslash frac\{dx\}\{(\backslash sin^\{n-2\}\; ax)(\backslash cos\; ax)\}\; \backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$
* $\backslash int\backslash frac\{\backslash sin\; ax\backslash ,dx\}\{\backslash cos^n\; ax\}\; =\; \backslash frac\{1\}\{a(n-1)\backslash cos^\{n-1\}\; ax\}\; +C\backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$
* $\backslash int\backslash frac\{\backslash sin^2\; ax\backslash ,dx\}\{\backslash cos\; ax\}\; =\; -\backslash frac\{1\}\{a\}\backslash sin\; ax+\backslash frac\{1\}\{a\}\backslash ln\backslash left|\backslash tan\backslash left(\backslash frac\{\backslash pi\}\{4\}+\backslash frac\{ax\}\{2\}\backslash right)\backslash right|+C$
* $\backslash int\backslash frac\{\backslash sin^2\; ax\backslash ,dx\}\{\backslash cos^n\; ax\}\; =\; \backslash frac\{\backslash sin\; ax\}\{a(n-1)\backslash cos^\{n-1\}ax\}-\backslash frac\{1\}\{n-1\}\backslash int\backslash frac\{dx\}\{\backslash cos^\{n-2\}ax\}\; \backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$
* $\backslash int\backslash frac\{\backslash sin^n\; ax\backslash ,dx\}\{\backslash cos\; ax\}\; =\; -\backslash frac\{\backslash sin^\{n-1\}\; ax\}\{a(n-1)\}\; +\; \backslash int\backslash frac\{\backslash sin^\{n-2\}\; ax\backslash ,dx\}\{\backslash cos\; ax\}\; \backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$
* $\backslash begin\{cases\}$
\int\frac{\sin^n ax\,dx}{\cos^m ax} = \frac{\sin^{n+1} ax}{a(m-1)\cos^{m-1} ax}-\frac{n-m+2}{m-1}\int\frac{\sin^n ax\,dx}{\cos^{m-2} ax} &\mbox{(for }m\neq 1\mbox{)} \\
\int\frac{\sin^n ax\,dx}{\cos^m ax} = \frac{\sin^{n-1} ax}{a(m-1)\cos^{m-1} ax}-\frac{n-1}{m-1}\int\frac{\sin^{n-2} ax\,dx}{\cos^{m-2} ax} &\mbox{(for }m\neq 1\mbox{)} \\
\int\frac{\sin^n ax\,dx}{\cos^m ax} = -\frac{\sin^{n-1} ax}{a(n-m)\cos^{m-1} ax}+\frac{n-1}{n-m}\int\frac{\sin^{n-2} ax\,dx}{\cos^m ax} &\mbox{(for }m\neq n\mbox{)}
\end{cases}
* $\backslash int\backslash frac\{\backslash cos\; ax\backslash ,dx\}\{\backslash sin^n\; ax\}\; =\; -\backslash frac\{1\}\{a(n-1)\backslash sin^\{n-1\}\; ax\}\; +C\backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$
* $\backslash int\backslash frac\{\backslash cos^2\; ax\backslash ,dx\}\{\backslash sin\; ax\}\; =\; \backslash frac\{1\}\{a\}\backslash left(\backslash cos\; ax+\backslash ln\backslash left|\backslash tan\backslash frac\{ax\}\{2\}\backslash right|\backslash right)\; +C$
* $\backslash int\backslash frac\{\backslash cos^2\; ax\backslash ,dx\}\{\backslash sin^n\; ax\}\; =\; -\backslash frac\{1\}\{n-1\}\backslash left(\backslash frac\{\backslash cos\; ax\}\{a\backslash sin^\{n-1\}\; ax\}+\backslash int\backslash frac\{dx\}\{\backslash sin^\{n-2\}\; ax\}\backslash right)\; \backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$
* $\backslash begin\{cases\}$
\int\frac{\cos^n ax\,dx}{\sin^m ax} = -\frac{\cos^{n+1} ax}{a(m-1)\sin^{m-1} ax} - \frac{n-m+2}{m-1}\int\frac{\cos^n ax\,dx}{\sin^{m-2} ax} &\mbox{(for }m\neq 1\mbox{)} \\
\int\frac{\cos^n ax\,dx}{\sin^m ax} = -\frac{\cos^{n-1} ax}{a(m-1)\sin^{m-1} ax} - \frac{n-1}{m-1}\int\frac{\cos^{n-2} ax\,dx}{\sin^{m-2} ax} &\mbox{(for }m\neq 1\mbox{)} \\
\int\frac{\cos^n ax\,dx}{\sin^m ax} = \frac{\cos^{n-1} ax}{a(n-m)\sin^{m-1} ax} + \frac{n-1}{n-m}\int\frac{\cos^{n-2} ax\,dx}{\sin^m ax} &\mbox{(for }m\neq n\mbox{)}
\end{cases}

8.1. 일반적인 형태의 적분

사인과 코사인의 합, 차, 곱 등으로 이루어진 함수들의 적분 공식을 나열한다. Bioche의 규칙을 사용하여 사인과 코사인의 유리 함수인 적분을 평가할 수 있다.

*$\backslash int\backslash frac\{dx\}\{\backslash cos\; ax\backslash pm\backslash sin\; ax\}\; =\; \backslash frac\{1\}\{a\backslash sqrt\{2\}\}\backslash ln\backslash left|\backslash tan\backslash left(\backslash frac\{ax\}\{2\}\backslash pm\backslash frac\{\backslash pi\}\{8\}\backslash right)\backslash right|+C$

*$\backslash int\backslash frac\{dx\}\{(\backslash cos\; ax\backslash pm\backslash sin\; ax)^2\}\; =\; \backslash frac\{1\}\{2a\}\backslash tan\backslash left(ax\backslash mp\backslash frac\{\backslash pi\}\{4\}\backslash right)+C$

*$\backslash int\backslash frac\{dx\}\{(\backslash cos\; x\; +\; \backslash sin\; x)^n\}\; =\; \backslash frac\{1\}\{n-1\}\backslash left(\backslash frac\{\backslash sin\; x\; -\; \backslash cos\; x\}\{(\backslash cos\; x\; +\; \backslash sin\; x)^\{n\; -\; 1\}\}\; -\; 2(n\; -\; 2)\backslash int\backslash frac\{dx\}\{(\backslash cos\; x\; +\; \backslash sin\; x)^\{n-2\}\}\; \backslash right)$

*$\backslash int\backslash frac\{\backslash cos\; ax\backslash ,dx\}\{\backslash cos\; ax\; +\; \backslash sin\; ax\}\; =\; \backslash frac\{x\}\{2\}\; +\; \backslash frac\{1\}\{2a\}\backslash ln\backslash left|\backslash sin\; ax\; +\; \backslash cos\; ax\backslash right|+C$

*$\backslash int\backslash frac\{\backslash cos\; ax\backslash ,dx\}\{\backslash cos\; ax\; -\; \backslash sin\; ax\}\; =\; \backslash frac\{x\}\{2\}\; -\; \backslash frac\{1\}\{2a\}\backslash ln\backslash left|\backslash sin\; ax\; -\; \backslash cos\; ax\backslash right|+C$

*$\backslash int\backslash frac\{\backslash sin\; ax\backslash ,dx\}\{\backslash cos\; ax\; +\; \backslash sin\; ax\}\; =\; \backslash frac\{x\}\{2\}\; -\; \backslash frac\{1\}\{2a\}\backslash ln\backslash left|\backslash sin\; ax\; +\; \backslash cos\; ax\backslash right|+C$

*$\backslash int\backslash frac\{\backslash sin\; ax\backslash ,dx\}\{\backslash cos\; ax\; -\; \backslash sin\; ax\}\; =\; -\backslash frac\{x\}\{2\}\; -\; \backslash frac\{1\}\{2a\}\backslash ln\backslash left|\backslash sin\; ax\; -\; \backslash cos\; ax\backslash right|+C$

*$\backslash int\backslash frac\{\backslash cos\; ax\backslash ,dx\}\{(\backslash sin\; ax)(1+\backslash cos\; ax)\}\; =\; -\backslash frac\{1\}\{4a\}\backslash tan^2\backslash frac\{ax\}\{2\}+\backslash frac\{1\}\{2a\}\backslash ln\backslash left|\backslash tan\backslash frac\{ax\}\{2\}\backslash right|+C$

*$\backslash int\backslash frac\{\backslash cos\; ax\backslash ,dx\}\{(\backslash sin\; ax)(1-\backslash cos\; ax)\}\; =\; -\backslash frac\{1\}\{4a\}\backslash cot^2\backslash frac\{ax\}\{2\}-\backslash frac\{1\}\{2a\}\backslash ln\backslash left|\backslash tan\backslash frac\{ax\}\{2\}\backslash right|+C$

*$\backslash int\backslash frac\{\backslash sin\; ax\backslash ,dx\}\{(\backslash cos\; ax)(1+\backslash sin\; ax)\}\; =\; \backslash frac\{1\}\{4a\}\backslash cot^2\backslash left(\backslash frac\{ax\}\{2\}+\backslash frac\{\backslash pi\}\{4\}\backslash right)+\backslash frac\{1\}\{2a\}\backslash ln\backslash left|\backslash tan\backslash left(\backslash frac\{ax\}\{2\}+\backslash frac\{\backslash pi\}\{4\}\backslash right)\backslash right|+C$

*$\backslash int\backslash frac\{\backslash sin\; ax\backslash ,dx\}\{(\backslash cos\; ax)(1-\backslash sin\; ax)\}\; =\; \backslash frac\{1\}\{4a\}\backslash tan^2\backslash left(\backslash frac\{ax\}\{2\}+\backslash frac\{\backslash pi\}\{4\}\backslash right)-\backslash frac\{1\}\{2a\}\backslash ln\backslash left|\backslash tan\backslash left(\backslash frac\{ax\}\{2\}+\backslash frac\{\backslash pi\}\{4\}\backslash right)\backslash right|+C$

*$\backslash int(\backslash sin\; ax)(\backslash cos\; ax)\backslash ,dx\; =\; \backslash frac\{1\}\{2a\}\backslash sin^2\; ax\; +C$

*$\backslash int(\backslash sin\; a\_1x)(\backslash cos\; a\_2x)\backslash ,dx\; =\; -\backslash frac\{\backslash cos((a\_1-a\_2)x)\}\{2(a\_1-a\_2)\}\; -\backslash frac\{\backslash cos((a\_1+a\_2)x)\}\{2(a\_1+a\_2)\}\; +C\backslash qquad\backslash mbox\{(for\; \}|a\_1|\backslash neq|a\_2|\backslash mbox\{)\}$

*$\backslash int(\backslash sin^n\; ax)(\backslash cos\; ax)\backslash ,dx\; =\; \backslash frac\{1\}\{a(n+1)\}\backslash sin^\{n+1\}\; ax\; +C\backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; -1\backslash mbox\{)\}$

*$\backslash int(\backslash sin\; ax)(\backslash cos^n\; ax)\backslash ,dx\; =\; -\backslash frac\{1\}\{a(n+1)\}\backslash cos^\{n+1\}\; ax\; +C\backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; -1\backslash mbox\{)\}$

*$\backslash begin\{align\}$
\int(\sin^n ax)(\cos^m ax)\,dx &= -\frac{(\sin^{n-1} ax)(\cos^{m+1} ax)}{a(n+m)}+\frac{n-1}{n+m}\int(\sin^{n-2} ax)(\cos^m ax)\,dx \qquad\mbox{(for }m,n>0\mbox{)} \\
&= \frac{(\sin^{n+1} ax)(\cos^{m-1} ax)}{a(n+m)} + \frac{m-1}{n+m}\int(\sin^n ax)(\cos^{m-2} ax)\,dx \qquad\mbox{(for }m,n>0\mbox{)}
\end{align}

*$\backslash int\backslash frac\{dx\}\{(\backslash sin\; ax)(\backslash cos\; ax)\}\; =\; \backslash frac\{1\}\{a\}\backslash ln\backslash left|\backslash tan\; ax\backslash right|+C$

*$\backslash int\backslash frac\{dx\}\{(\backslash sin\; ax)(\backslash cos^n\; ax)\}\; =\; \backslash frac\{1\}\{a(n-1)\backslash cos^\{n-1\}\; ax\}+\backslash int\backslash frac\{dx\}\{(\backslash sin\; ax)(\backslash cos^\{n-2\}\; ax)\}\; \backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$

*$\backslash int\backslash frac\{dx\}\{(\backslash sin^n\; ax)(\backslash cos\; ax)\}\; =\; -\backslash frac\{1\}\{a(n-1)\backslash sin^\{n-1\}\; ax\}+\backslash int\backslash frac\{dx\}\{(\backslash sin^\{n-2\}\; ax)(\backslash cos\; ax)\}\; \backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$

*$\backslash int\backslash frac\{\backslash sin\; ax\backslash ,dx\}\{\backslash cos^n\; ax\}\; =\; \backslash frac\{1\}\{a(n-1)\backslash cos^\{n-1\}\; ax\}\; +C\backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$

*$\backslash int\backslash frac\{\backslash sin^2\; ax\backslash ,dx\}\{\backslash cos\; ax\}\; =\; -\backslash frac\{1\}\{a\}\backslash sin\; ax+\backslash frac\{1\}\{a\}\backslash ln\backslash left|\backslash tan\backslash left(\backslash frac\{\backslash pi\}\{4\}+\backslash frac\{ax\}\{2\}\backslash right)\backslash right|+C$

*$\backslash int\backslash frac\{\backslash sin^2\; ax\backslash ,dx\}\{\backslash cos^n\; ax\}\; =\; \backslash frac\{\backslash sin\; ax\}\{a(n-1)\backslash cos^\{n-1\}ax\}-\backslash frac\{1\}\{n-1\}\backslash int\backslash frac\{dx\}\{\backslash cos^\{n-2\}ax\}\; \backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$

*$\backslash begin\{align\}$
\int \frac{\sin^2 x}{1 + \cos^2 x} \, dx &= \sqrt{2}\operatorname{arctangant}\left(\frac{\tan x}{\sqrt{2}}\right) - x \qquad\mbox{(for x in}] - \frac{\pi}{2} ; + \frac{\pi}{2} [\mbox{)} \\
&= \sqrt{2}\operatorname{arctangant}\left(\frac{\tan x}{\sqrt{2}}\right)-\operatorname{arctangant}\left(\tan x\right) \qquad\mbox{(this time x being any real number }\mbox{)}
\end{align}

*$\backslash int\backslash frac\{\backslash sin^n\; ax\backslash ,dx\}\{\backslash cos\; ax\}\; =\; -\backslash frac\{\backslash sin^\{n-1\}\; ax\}\{a(n-1)\}\; +\; \backslash int\backslash frac\{\backslash sin^\{n-2\}\; ax\backslash ,dx\}\{\backslash cos\; ax\}\; \backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$

*$\backslash int\backslash frac\{\backslash sin^n\; ax\backslash ,dx\}\{\backslash cos^m\; ax\}\; =\; \backslash begin\{cases\}$
\frac{\sin^{n+1} ax}{a(m-1)\cos^{m-1} ax}-\frac{n-m+2}{m-1}\int\frac{\sin^n ax\,dx}{\cos^{m-2} ax} &\mbox{(for }m\neq 1\mbox{)} \\
\frac{\sin^{n-1} ax}{a(m-1)\cos^{m-1} ax}-\frac{n-1}{m-1}\int\frac{\sin^{n-2} ax\,dx}{\cos^{m-2} ax} &\mbox{(for }m\neq 1\mbox{)} \\
-\frac{\sin^{n-1} ax}{a(n-m)\cos^{m-1} ax}+\frac{n-1}{n-m}\int\frac{\sin^{n-2} ax\,dx}{\cos^m ax} &\mbox{(for }m\neq n\mbox{)}
\end{cases}

*$\backslash int\backslash frac\{\backslash cos\; ax\backslash ,dx\}\{\backslash sin^n\; ax\}\; =\; -\backslash frac\{1\}\{a(n-1)\backslash sin^\{n-1\}\; ax\}\; +C\backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$

*$\backslash int\backslash frac\{\backslash cos^2\; ax\backslash ,dx\}\{\backslash sin\; ax\}\; =\; \backslash frac\{1\}\{a\}\backslash left(\backslash cos\; ax+\backslash ln\backslash left|\backslash tan\backslash frac\{ax\}\{2\}\backslash right|\backslash right)\; +C$

*$\backslash int\backslash frac\{\backslash cos^2\; ax\backslash ,dx\}\{\backslash sin^n\; ax\}\; =\; -\backslash frac\{1\}\{n-1\}\backslash left(\backslash frac\{\backslash cos\; ax\}\{a\backslash sin^\{n-1\}\; ax\}+\backslash int\backslash frac\{dx\}\{\backslash sin^\{n-2\}\; ax\}\backslash right)\; \backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$

*$\backslash int\backslash frac\{\backslash cos^n\; ax\backslash ,dx\}\{\backslash sin^m\; ax\}\; =\; \backslash begin\{cases\}$
-\frac{\cos^{n+1} ax}{a(m-1)\sin^{m-1} ax} - \frac{n-m+2}{m-1}\int\frac{\cos^n ax\,dx}{\sin^{m-2} ax} &\mbox{(for }m\neq 1\mbox{)} \\
-\frac{\cos^{n-1} ax}{a(m-1)\sin^{m-1} ax} - \frac{n-1}{m-1}\int\frac{\cos^{n-2} ax\,dx}{\sin^{m-2} ax} &\mbox{(for }m\neq 1\mbox{)} \\
\frac{\cos^{n-1} ax}{a(n-m)\sin^{m-1} ax} + \frac{n-1}{n-m}\int\frac{\cos^{n-2} ax\,dx}{\sin^m ax} &\mbox{(for }m\neq n\mbox{)}
\end{cases}

8.2. 분수 형태의 적분

사인과 코사인을 포함하는 분수 형태의 함수에 대한 적분 공식은 다음과 같다. Bioche의 규칙을 사용하여 적분을 계산할 수 있다.

*$\backslash int\backslash frac\{dx\}\{\backslash cos\; ax\backslash pm\backslash sin\; ax\}\; =\; \backslash frac\{1\}\{a\backslash sqrt\{2\}\}\backslash ln\backslash left|\backslash tan\backslash left(\backslash frac\{ax\}\{2\}\backslash pm\backslash frac\{\backslash pi\}\{8\}\backslash right)\backslash right|+C$

*$\backslash int\backslash frac\{dx\}\{(\backslash cos\; ax\backslash pm\backslash sin\; ax)^2\}\; =\; \backslash frac\{1\}\{2a\}\backslash tan\backslash left(ax\backslash mp\backslash frac\{\backslash pi\}\{4\}\backslash right)+C$

*$\backslash int\backslash frac\{\backslash cos\; ax\backslash ,dx\}\{\backslash cos\; ax\; +\; \backslash sin\; ax\}\; =\; \backslash frac\{x\}\{2\}\; +\; \backslash frac\{1\}\{2a\}\backslash ln\backslash left|\backslash sin\; ax\; +\; \backslash cos\; ax\backslash right|+C$

*$\backslash int\backslash frac\{\backslash cos\; ax\backslash ,dx\}\{\backslash cos\; ax\; -\; \backslash sin\; ax\}\; =\; \backslash frac\{x\}\{2\}\; -\; \backslash frac\{1\}\{2a\}\backslash ln\backslash left|\backslash sin\; ax\; -\; \backslash cos\; ax\backslash right|+C$

*$\backslash int\backslash frac\{\backslash sin\; ax\backslash ,dx\}\{\backslash cos\; ax\; +\; \backslash sin\; ax\}\; =\; \backslash frac\{x\}\{2\}\; -\; \backslash frac\{1\}\{2a\}\backslash ln\backslash left|\backslash sin\; ax\; +\; \backslash cos\; ax\backslash right|+C$

*$\backslash int\backslash frac\{\backslash sin\; ax\backslash ,dx\}\{\backslash cos\; ax\; -\; \backslash sin\; ax\}\; =\; -\backslash frac\{x\}\{2\}\; -\; \backslash frac\{1\}\{2a\}\backslash ln\backslash left|\backslash sin\; ax\; -\; \backslash cos\; ax\backslash right|+C$

*$\backslash int\backslash frac\{\backslash cos\; ax\backslash ,dx\}\{(\backslash sin\; ax)(1+\backslash cos\; ax)\}\; =\; -\backslash frac\{1\}\{4a\}\backslash tan^2\backslash frac\{ax\}\{2\}+\backslash frac\{1\}\{2a\}\backslash ln\backslash left|\backslash tan\backslash frac\{ax\}\{2\}\backslash right|+C$

*$\backslash int\backslash frac\{\backslash cos\; ax\backslash ,dx\}\{(\backslash sin\; ax)(1-\backslash cos\; ax)\}\; =\; -\backslash frac\{1\}\{4a\}\backslash cot^2\backslash frac\{ax\}\{2\}-\backslash frac\{1\}\{2a\}\backslash ln\backslash left|\backslash tan\backslash frac\{ax\}\{2\}\backslash right|+C$

*$\backslash int\backslash frac\{\backslash sin\; ax\backslash ,dx\}\{(\backslash cos\; ax)(1+\backslash sin\; ax)\}\; =\; \backslash frac\{1\}\{4a\}\backslash cot^2\backslash left(\backslash frac\{ax\}\{2\}+\backslash frac\{\backslash pi\}\{4\}\backslash right)+\backslash frac\{1\}\{2a\}\backslash ln\backslash left|\backslash tan\backslash left(\backslash frac\{ax\}\{2\}+\backslash frac\{\backslash pi\}\{4\}\backslash right)\backslash right|+C$

*$\backslash int\backslash frac\{\backslash sin\; ax\backslash ,dx\}\{(\backslash cos\; ax)(1-\backslash sin\; ax)\}\; =\; \backslash frac\{1\}\{4a\}\backslash tan^2\backslash left(\backslash frac\{ax\}\{2\}+\backslash frac\{\backslash pi\}\{4\}\backslash right)-\backslash frac\{1\}\{2a\}\backslash ln\backslash left|\backslash tan\backslash left(\backslash frac\{ax\}\{2\}+\backslash frac\{\backslash pi\}\{4\}\backslash right)\backslash right|+C$

*$\backslash int\backslash frac\{dx\}\{(\backslash sin\; ax)(\backslash cos\; ax)\}\; =\; \backslash frac\{1\}\{a\}\backslash ln\backslash left|\backslash tan\; ax\backslash right|+C$

*$\backslash int\backslash frac\{\backslash sin^2\; ax\backslash ,dx\}\{\backslash cos\; ax\}\; =\; -\backslash frac\{1\}\{a\}\backslash sin\; ax+\backslash frac\{1\}\{a\}\backslash ln\backslash left|\backslash tan\backslash left(\backslash frac\{\backslash pi\}\{4\}+\backslash frac\{ax\}\{2\}\backslash right)\backslash right|+C$

*$\backslash int\backslash frac\{\backslash cos^2\; ax\backslash ,dx\}\{\backslash sin\; ax\}\; =\; \backslash frac\{1\}\{a\}\backslash left(\backslash cos\; ax+\backslash ln\backslash left|\backslash tan\backslash frac\{ax\}\{2\}\backslash right|\backslash right)\; +C$

* 분모 또는 분자에 사인과 코사인 함수의 거듭제곱을 포함하고 있는 경우의 적분 공식은 다음과 같다.

*$\backslash int\backslash frac\{dx\}\{(\backslash cos\; x\; +\; \backslash sin\; x)^n\}\; =\; \backslash frac\{1\}\{n-1\}\backslash left(\backslash frac\{\backslash sin\; x\; -\; \backslash cos\; x\}\{(\backslash cos\; x\; +\; \backslash sin\; x)^\{n\; -\; 1\}\}\; -\; 2(n\; -\; 2)\backslash int\backslash frac\{dx\}\{(\backslash cos\; x\; +\; \backslash sin\; x)^\{n-2\}\}\; \backslash right)$

*$\backslash int(\backslash sin\; a\_1x)(\backslash cos\; a\_2x)\backslash ,dx\; =\; -\backslash frac\{\backslash cos((a\_1-a\_2)x)\}\{2(a\_1-a\_2)\}\; -\backslash frac\{\backslash cos((a\_1+a\_2)x)\}\{2(a\_1+a\_2)\}\; +C\backslash qquad\backslash mbox\{(for\; \}|a\_1|\backslash neq|a\_2|\backslash mbox\{)\}$

*$\backslash int\backslash frac\{dx\}\{(\backslash sin\; ax)(\backslash cos^n\; ax)\}\; =\; \backslash frac\{1\}\{a(n-1)\backslash cos^\{n-1\}\; ax\}+\backslash int\backslash frac\{dx\}\{(\backslash sin\; ax)(\backslash cos^\{n-2\}\; ax)\}\; \backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$

*$\backslash int\backslash frac\{dx\}\{(\backslash sin^n\; ax)(\backslash cos\; ax)\}\; =\; -\backslash frac\{1\}\{a(n-1)\backslash sin^\{n-1\}\; ax\}+\backslash int\backslash frac\{dx\}\{(\backslash sin^\{n-2\}\; ax)(\backslash cos\; ax)\}\; \backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$

*$\backslash int\backslash frac\{\backslash sin\; ax\backslash ,dx\}\{\backslash cos^n\; ax\}\; =\; \backslash frac\{1\}\{a(n-1)\backslash cos^\{n-1\}\; ax\}\; +C\backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$

*$\backslash int\backslash frac\{\backslash sin^2\; ax\backslash ,dx\}\{\backslash cos^n\; ax\}\; =\; \backslash frac\{\backslash sin\; ax\}\{a(n-1)\backslash cos^\{n-1\}ax\}-\backslash frac\{1\}\{n-1\}\backslash int\backslash frac\{dx\}\{\backslash cos^\{n-2\}ax\}\; \backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$

*$\backslash int\backslash frac\{\backslash sin^n\; ax\backslash ,dx\}\{\backslash cos\; ax\}\; =\; -\backslash frac\{\backslash sin^\{n-1\}\; ax\}\{a(n-1)\}\; +\; \backslash int\backslash frac\{\backslash sin^\{n-2\}\; ax\backslash ,dx\}\{\backslash cos\; ax\}\; \backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$

*$\backslash int\backslash frac\{\backslash cos\; ax\backslash ,dx\}\{\backslash sin^n\; ax\}\; =\; -\backslash frac\{1\}\{a(n-1)\backslash sin^\{n-1\}\; ax\}\; +C\backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$

*$\backslash int\backslash frac\{\backslash cos^2\; ax\backslash ,dx\}\{\backslash sin^n\; ax\}\; =\; -\backslash frac\{1\}\{n-1\}\backslash left(\backslash frac\{\backslash cos\; ax\}\{a\backslash sin^\{n-1\}\; ax\}+\backslash int\backslash frac\{dx\}\{\backslash sin^\{n-2\}\; ax\}\backslash right)\; \backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$

*$\backslash int\backslash frac\{\backslash sin^n\; ax\backslash ,dx\}\{\backslash cos^m\; ax\}\; =\; \backslash begin\{cases\}$
\frac{\sin^{n+1} ax}{a(m-1)\cos^{m-1} ax}-\frac{n-m+2}{m-1}\int\frac{\sin^n ax\,dx}{\cos^{m-2} ax} &\mbox{(for }m\neq 1\mbox{)} \\
\frac{\sin^{n-1} ax}{a(m-1)\cos^{m-1} ax}-\frac{n-1}{m-1}\int\frac{\sin^{n-2} ax\,dx}{\cos^{m-2} ax} &\mbox{(for }m\neq 1\mbox{)} \\
-\frac{\sin^{n-1} ax}{a(n-m)\cos^{m-1} ax}+\frac{n-1}{n-m}\int\frac{\sin^{n-2} ax\,dx}{\cos^m ax} &\mbox{(for }m\neq n\mbox{)}
\end{cases}

*$\backslash int\backslash frac\{\backslash cos^n\; ax\backslash ,dx\}\{\backslash sin^m\; ax\}\; =\; \backslash begin\{cases\}$
-\frac{\cos^{n+1} ax}{a(m-1)\sin^{m-1} ax} - \frac{n-m+2}{m-1}\int\frac{\cos^n ax\,dx}{\sin^{m-2} ax} &\mbox{(for }m\neq 1\mbox{)} \\
-\frac{\cos^{n-1} ax}{a(m-1)\sin^{m-1} ax} - \frac{n-1}{m-1}\int\frac{\cos^{n-2} ax\,dx}{\sin^{m-2} ax} &\mbox{(for }m\neq 1\mbox{)} \\
\frac{\cos^{n-1} ax}{a(n-m)\sin^{m-1} ax} + \frac{n-1}{n-m}\int\frac{\cos^{n-2} ax\,dx}{\sin^m ax} &\mbox{(for }m\neq n\mbox{)}
\end{cases}

9. 사인/코사인과 탄젠트/코탄젠트를 포함하는 함수의 적분

$\backslash int\; (\backslash sin\; ax)(\backslash tan\; ax)\backslash ,dx\; =\; \backslash frac\{1\}\{a\}(\backslash ln|\backslash sec\; ax\; +\; \backslash tan\; ax|\; -\; \backslash sin\; ax)+C$

$\backslash int\backslash frac\{\backslash tan^n\; ax\backslash ,dx\}\{\backslash sin^2\; ax\}\; =\; \backslash frac\{1\}\{a(n-1)\}\backslash tan^\{n-1\}\; (ax)\; +C\backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$

$\backslash int\backslash frac\{\backslash tan^n\; ax\backslash ,dx\}\{\backslash cos^2\; ax\}\; =\; \backslash frac\{1\}\{a(n+1)\}\backslash tan^\{n+1\}\; ax\; +C\backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; -1\backslash mbox\{)\}$

$\backslash int\backslash frac\{\backslash cot^n\; ax\backslash ,dx\}\{\backslash sin^2\; ax\}\; =\; -\backslash frac\{1\}\{a(n+1)\}\backslash cot^\{n+1\}\; ax\; +C\backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; -1\backslash mbox\{)\}$

$\backslash int\backslash frac\{\backslash cot^n\; ax\backslash ,dx\}\{\backslash cos^2\; ax\}\; =\; \backslash frac\{1\}\{a(1-n)\}\backslash tan^\{1-n\}\; ax\; +C\backslash qquad\backslash mbox\{(for\; \}n\backslash neq\; 1\backslash mbox\{)\}$

$\backslash int(\backslash sec\; x)(\backslash tan\; x)\backslash ,dx=\; \backslash sec\; x\; +\; C$

$\backslash int(\backslash csc\; x)(\backslash cot\; x)\backslash ,dx=\; -\backslash csc\; x\; +\; C$

$\backslash int\_^\}\; \backslash sin^n\; x\; \backslash ,\; dx\; =\; \backslash int\_^\}\; \backslash cos^n\; x\; \backslash ,\; dx\; =\; \backslash begin\{cases\}$
\frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdots \frac{3}{4} \cdot \frac{1}{2} \cdot \frac{\pi}{2}, & \text{if } n\text{ is even} \\
\frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdots \frac{4}{5} \cdot \frac{2}{3}, & \text{if } n\text{ is odd and more than 1} \\
1, & \text{if } n=1
\end{cases}

$\backslash int\_^\backslash sin^\{2m+1\}\{x\}\backslash cos^\{2n+1\}\{x\}\backslash ,dx\; =\; 0\; \backslash !\; \backslash qquad\; n,m\; \backslash in\; \backslash mathbb\{Z\}$

10. 대칭성을 이용한 정적분

적분 구간의 대칭성을 이용하면 삼각함수의 정적분 계산을 간단하게 할 수 있다.

*$\backslash int\_^\backslash sin\{x\}\backslash ,dx\; =\; 0$
*$\backslash int\_^\backslash cos\; \{x\}\backslash ,dx\; =\; 2\backslash int\_^\backslash cos\; \{x\}\backslash ,dx\; =\; 2\backslash int\_^\backslash cos\; \{x\}\backslash ,dx\; =\; 2\backslash sin\; \{c\}$
*$\backslash int\_^\backslash tan\; \{x\}\backslash ,dx\; =\; 0$
*$\backslash int\_\{-\backslash frac\{a\}\{2\}\}^\{\backslash frac\{a\}\{2\}\}\; x^2\backslash cos^2\; \{\backslash frac\{n\backslash pi\; x\}\{a\}\}\backslash ,dx\; =\; \backslash frac\{a^3(n^2\backslash pi^2-6)\}\{24n^2\backslash pi^2\}\; \backslash qquad\backslash mbox\{(for\; \}n=1,3,5...\backslash mbox\{)\}$
*$\backslash int\_\{\backslash frac\{-a\}\{2\}\}^\{\backslash frac\{a\}\{2\}\}\; x^2\backslash sin^2\; \{\backslash frac\{n\backslash pi\; x\}\{a\}\}\backslash ,dx\; =\; \backslash frac\{a^3(n^2\backslash pi^2-6(-1)^n)\}\{24n^2\backslash pi^2\}\; =\; \backslash frac\{a^3\}\{24\}\; (1-6\backslash frac\{(-1)^n\}\{n^2\backslash pi^2\})\; \backslash qquad\backslash mbox\{(for\; \}n=1,2,3,...\backslash mbox\{)\}$

11. 특정 구간에서의 적분

$\backslash int\_^\backslash sin\{x\}\backslash ,dx\; =\; 0$
$\backslash int\_^\backslash cos\; \{x\}\backslash ,dx\; =\; 2\backslash int\_^\backslash cos\; \{x\}\backslash ,dx\; =\; 2\backslash int\_^\backslash cos\; \{x\}\backslash ,dx\; =\; 2\backslash sin\; \{c\}$
$\backslash int\_^\backslash tan\; \{x\}\backslash ,dx\; =\; 0$
$\backslash int\_\{-\backslash frac\{a\}\{2\}\}^\{\backslash frac\{a\}\{2\}\}\; x^2\backslash cos^2\; \{\backslash frac\{n\backslash pi\; x\}\{a\}\}\backslash ,dx\; =\; \backslash frac\{a^3(n^2\backslash pi^2-6)\}\{24n^2\backslash pi^2\}\; \backslash qquad\backslash mbox\{(for\; \}n=1,3,5...\backslash mbox\{)\}$
$\backslash int\_\{\backslash frac\{-a\}\{2\}\}^\{\backslash frac\{a\}\{2\}\}\; x^2\backslash sin^2\; \{\backslash frac\{n\backslash pi\; x\}\{a\}\}\backslash ,dx\; =\; \backslash frac\{a^3(n^2\backslash pi^2-6(-1)^n)\}\{24n^2\backslash pi^2\}\; =\; \backslash frac\{a^3\}\{24\}\; (1-6\backslash frac\{(-1)^n\}\{n^2\backslash pi^2\})\; \backslash qquad\backslash mbox\{(for\; \}n=1,2,3,...\backslash mbox\{)\}$
베타 함수 $B(a,b)$를 사용하여 다음과 같이 쓸 수 있다.
* $\backslash int\_^\}\; \backslash sin^n\; x\; \backslash ,\; dx\; =\; \backslash int\_^\}\; \backslash cos^n\; x\; \backslash ,\; dx\; =\; \backslash frac\{1\}\{2\}\; B\backslash left(\; \backslash frac\{n+1\}\{2\},\; \backslash frac\{1\}\{2\}\backslash right)\; =\; \backslash begin\{cases\}$
\frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdots \frac{3}{4} \cdot \frac{1}{2} \cdot \frac{\pi}{2}, & \text{if } n\text{ is even} \\
\frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdots \frac{4}{5} \cdot \frac{2}{3}, & \text{if } n\text{ is odd and more than 1} \\
1, & \text{if } n=1
\end{cases}
* $\backslash int\_^\backslash sin^\{2m+1\}\{x\}\backslash cos^\{n\}\{x\}\backslash ,dx\; =\; 0\; \backslash !\; \backslash qquad\; n,m\; \backslash in\; \backslash mathbb\{Z\}$
* $\backslash int\_^\backslash sin^\{m\}\{x\}\backslash cos^\{2n+1\}\{x\}\backslash ,dx\; =\; 0\; \backslash !\; \backslash qquad\; n,m\; \backslash in\; \backslash mathbb\{Z\}$