다음은 삼각 함수(trigonometric functions)의 적분(integral) (역도함수(antiderivative))의 목록입니다. 지수 함수와 삼각 함수 둘 다를 포함하는 역도함수에 대해, 지수 함수의 적분의 목록(List of integrals of exponential functions)을 참조하십시오. 역도함수의 전체 목록에 대해, 적분의 목록(Lists of integrals)을 참조하십시오. 삼각 함수를 포함하는 특수 역도함수에 대해, 삼각 적분(Trigonometric integral)을 참조하십시오.
일반적으로, 만약 함수 \(\sin(x)\)가 임의의 삼각 함수이고, \(\cos(x)\)가 그것의 도함수이면, 다음입니다:
\(\displaystyle \int a\cos nx\,dx = \frac{a}{n}\sin nx+C\)
모든 공식에서, 상수 a는 비-영으로 가정되고, C는 적분의 상수(constant of integration)를 표시합니다.
Integrands involving only sine
\(\displaystyle \int\sin ax\,dx = -\frac{1}{a}\cos ax+C\)
\(\displaystyle \int\sin^2 {ax}\,dx = \frac{x}{2} - \frac{1}{4a} \sin 2ax +C= \frac{x}{2} - \frac{1}{2a} \sin ax\cos ax +C\)
\(\displaystyle \int\sin^3 {ax}\,dx = \frac{\cos 3ax}{12a} - \frac{3 \cos ax}{4a} +C\)
\(\displaystyle \int x\sin^2 {ax}\,dx = \frac{x^2}{4} - \frac{x}{4a} \sin 2ax - \frac{1}{8a^2} \cos 2ax +C\)
\(\displaystyle \int x^2\sin^2 {ax}\,dx = \frac{x^3}{6} - \left( \frac {x^2}{4a} - \frac{1}{8a^3} \right) \sin 2ax - \frac{x}{4a^2} \cos 2ax +C\)
\(\displaystyle \int x\sin ax\,dx = \frac{\sin ax}{a^2}-\frac{x\cos ax}{a}+C\)
\(\displaystyle \int(\sin b_1x)(\sin b_2x)\,dx = \frac{\sin((b_2-b_1)x)}{2(b_2-b_1)}-\frac{\sin((b_1+b_2)x)}{2(b_1+b_2)}+C \qquad\mbox{(for }|b_1|\neq|b_2|\mbox{)}\)
\(\displaystyle \int\sin^n {ax}\,dx = -\frac{\sin^{n-1} ax\cos ax}{na} + \frac{n-1}{n}\int\sin^{n-2} ax\,dx \qquad\mbox{(for }n>0\mbox{)}\)
\(\displaystyle \int\frac{dx}{\sin ax} = -\frac{1}{a}\ln{\left| \csc{ax}+\cot{ax}\right|}+C\)
\(\displaystyle \int\frac{dx}{\sin^n ax} = \frac{\cos ax}{a(1-n) \sin^{n-1} ax}+\frac{n-2}{n-1}\int\frac{dx}{\sin^{n-2}ax} \qquad\mbox{(for }n>1\mbox{)}\)
\(\displaystyle \begin{align}
\int x^n\sin ax\,dx &= -\frac{x^n}{a}\cos ax+\frac{n}{a}\int x^{n-1}\cos ax\,dx \\
&= \sum_{k=0}^{2k\leq n} (-1)^{k+1} \frac{x^{n-2k}}{a^{1+2k}}\frac{n!}{(n-2k)!} \cos ax +\sum_{k=0}^{2k+1\leq n}(-1)^k \frac{x^{n-1-2k}}{a^{2+2k}}\frac{n!}{(n-2k-1)!} \sin ax \\
&= - \sum_{k=0}^n \frac{x^{n-k}}{a^{1+k}}\frac{n!}{(n-k)!}\cos\left(ax+k\frac{\pi}{2}\right) \qquad\mbox{(for }n>0\mbox{)}
\end{align}\)
\(\displaystyle \int\frac{\sin ax}{x}\,dx = \sum_{n=0}^\infty (-1)^n\frac{(ax)^{2n+1}}{(2n+1)\cdot (2n+1)!} +C\)
\(\displaystyle \int\frac{\sin ax}{x^n}\,dx = -\frac{\sin ax}{(n-1)x^{n-1}} + \frac{a}{n-1}\int\frac{\cos ax}{x^{n-1}}\,dx\)
\(\displaystyle \int\frac{dx}{1\pm\sin ax} = \frac{1}{a}\tan\left(\frac{ax}{2}\mp\frac{\pi}{4}\right)+C\)
\(\displaystyle \int\frac{x\,dx}{1+\sin ax} = \frac{x}{a}\tan\left(\frac{ax}{2} - \frac{\pi}{4}\right)+\frac{2}{a^2}\ln\left|\cos\left(\frac{ax}{2}-\frac{\pi}{4}\right)\right|+C\)
\(\displaystyle \int\frac{x\,dx}{1-\sin ax} = \frac{x}{a}\cot\left(\frac{\pi}{4} - \frac{ax}{2}\right)+\frac{2}{a^2}\ln\left|\sin\left(\frac{\pi}{4}-\frac{ax}{2}\right)\right|+C\)
\(\displaystyle \int\frac{\sin ax\,dx}{1\pm\sin ax} = \pm x+\frac{1}{a}\tan\left(\frac{\pi}{4}\mp\frac{ax}{2}\right)+C\)
Integrands involving only cosine
\(\displaystyle \int\cos ax\,dx = \frac{1}{a}\sin ax+C\)
\(\displaystyle \int\cos^2 {ax}\,dx = \frac{x}{2} + \frac{1}{4a} \sin 2ax +C = \frac{x}{2} + \frac{1}{2a} \sin ax\cos ax +C\)
\(\displaystyle \int\cos^n ax\,dx = \frac{\cos^{n-1} ax\sin ax}{na} + \frac{n-1}{n}\int\cos^{n-2} ax\,dx \qquad\mbox{(for }n>0\mbox{)}\)
\(\displaystyle \int x\cos ax\,dx = \frac{\cos ax}{a^2} + \frac{x\sin ax}{a}+C\)
\(\displaystyle \int x^2\cos^2 {ax}\,dx = \frac{x^3}{6} + \left( \frac {x^2}{4a} - \frac{1}{8a^3} \right) \sin 2ax + \frac{x}{4a^2} \cos 2ax +C\)
\(\displaystyle \begin{align}
\int x^n\cos ax\,dx &= \frac{x^n\sin ax}{a} - \frac{n}{a}\int x^{n-1}\sin ax\,dx \\
&= \sum_{k=0}^{2k+1\leq n} (-1)^{k} \frac{x^{n-2k-1}}{a^{2+2k}}\frac{n!}{(n-2k-1)!} \cos ax +\sum_{k=0}^{2k\leq n}(-1)^{k} \frac{x^{n-2k}}{a^{1+2k}}\frac{n!}{(n-2k)!} \sin ax \\
&=\sum_{k=0}^n (-1)^{\lfloor k/2 \rfloor} \frac{x^{n-k}}{a^{1+k}}\frac{n!}{(n-k)!}\cos\left(ax -\frac{(-1)^k+1}{2}\frac{\pi}{2}\right) \\
&=\sum_{k=0}^n \frac{x^{n-k}}{a^{1+k}}\frac{n!}{(n-k)!}\sin\left(ax+k\frac{\pi}{2}\right) \qquad\mbox{(for }n>0\mbox{)}
\end{align}\)
\(\displaystyle \int\frac{\cos ax}{x}\,dx = \ln|ax|+\sum_{k=1}^\infty (-1)^k\frac{(ax)^{2k}}{2k\cdot(2k)!}+C\)
\(\displaystyle \int\frac{\cos ax}{x^n}\,dx = -\frac{\cos ax}{(n-1)x^{n-1}}-\frac{a}{n-1}\int\frac{\sin ax}{x^{n-1}}\,dx \qquad\mbox{(for }n\neq 1\mbox{)}\)
\(\displaystyle \int\frac{dx}{\cos ax} = \frac{1}{a}\ln\left|\tan\left(\frac{ax}{2}+\frac{\pi}{4}\right)\right|+C\)
\(\displaystyle \int\frac{dx}{\cos^n ax} = \frac{\sin ax}{a(n-1) \cos^{n-1} ax} + \frac{n-2}{n-1}\int\frac{dx}{\cos^{n-2} ax} \qquad\mbox{(for }n>1\mbox{)}\)
\(\displaystyle \int\frac{dx}{1+\cos ax} = \frac{1}{a}\tan\frac{ax}{2}+C\)
\(\displaystyle \int\frac{dx}{1-\cos ax} = -\frac{1}{a}\cot\frac{ax}{2}+C\)
\(\displaystyle \int\frac{x\,dx}{1+\cos ax} = \frac{x}{a}\tan\frac{ax}{2} + \frac{2}{a^2}\ln\left|\cos\frac{ax}{2}\right|+C\)
\(\displaystyle \int\frac{x\,dx}{1-\cos ax} = -\frac{x}{a}\cot\frac{ax}{2}+\frac{2}{a^2}\ln\left|\sin\frac{ax}{2}\right|+C\)
\(\displaystyle \int\frac{\cos ax\,dx}{1+\cos ax} = x - \frac{1}{a}\tan\frac{ax}{2}+C\)
\(\displaystyle \int\frac{\cos ax\,dx}{1-\cos ax} = -x-\frac{1}{a}\cot\frac{ax}{2}+C\)
\(\displaystyle \int(\cos a_1x)(\cos a_2x)\,dx = \frac{\sin((a_2-a_1)x)}{2(a_2-a_1)}+\frac{\sin((a_2+a_1)x)}{2(a_2+a_1)}+C \qquad\mbox{(for }|a_1|\neq|a_2|\mbox{)}\)
Integrands involving only tangent
\(\displaystyle \int\tan ax\,dx = -\frac{1}{a}\ln|\cos ax|+C = \frac{1}{a}\ln|\sec ax|+C\)
\(\displaystyle \int \tan^2{x} \, dx = \tan{x} - x +C\)
\(\displaystyle \int\tan^n ax\,dx = \frac{1}{a(n-1)}\tan^{n-1} ax-\int\tan^{n-2} ax\,dx \qquad\mbox{(for }n\neq 1\mbox{)}\)
\(\displaystyle \int\frac{dx}{q \tan ax + p} = \frac{1}{p^2 + q^2}(px + \frac{q}{a}\ln|q\sin ax + p\cos ax|)+C \qquad\mbox{(for }p^2 + q^2\neq 0\mbox{)}\)
\(\displaystyle \int\frac{dx}{\tan ax \pm 1} = \pm \frac{x}{2} + \frac{1}{2a}\ln|\sin ax \pm \cos ax|+C\)
\(\displaystyle \int\frac{\tan ax\,dx}{\tan ax \pm 1} = \frac{x}{2} \mp \frac{1}{2a}\ln|\sin ax \pm \cos ax|+C\)
Integrands involving only secant
\(\displaystyle \int \sec{ax} \, dx = \frac{1}{a}\ln{\left| \sec{ax} + \tan{ax}\right|}+C= \frac{1}{a}\ln{\left| \tan{\left(\frac{ax}{2} + \frac{\pi}{4} \right)}\right|}+C\)
\(\displaystyle \int \sec^2{x} \, dx = \tan{x}+C\)
\(\displaystyle \int \sec^3{x} \, dx = \frac{1}{2}\sec x \tan x + \frac{1}{2}\ln|\sec x + \tan x| + C.\)
\(\displaystyle \int \sec^n{ax} \, dx = \frac{\sec^{n-2}{ax} \tan {ax}}{a(n-1)} \,+\, \frac{n-2}{n-1}\int \sec^{n-2}{ax} \, dx \qquad \mbox{ (for }n \ne 1\mbox{)}\)
\(\displaystyle \int \frac{dx}{\sec{x} + 1} = x - \tan{\frac{x}{2}}+C\)
\(\displaystyle \int \frac{dx}{\sec{x} - 1} = - x - \cot{\frac{x}{2}}+C\)
Integrands involving only cosecant
\(\displaystyle \int \csc{ax} \, dx= -\frac{1}{a}\ln{\left| \csc{ax}+\cot{ax}\right|}+C= \frac{1}{a}\ln{\left| \csc{ax}-\cot{ax}\right|}+C = \frac{1}{a}\ln{\left| \tan{\left( \frac{ax}{2} \right)}\right|}+C\)
\(\displaystyle \int \csc^2{x} \, dx = -\cot{x}+C\)
\(\displaystyle \int \csc^3{x} \, dx = -\frac{1}{2}\csc x \cot x - \frac{1}{2}\ln|\csc x + \cot x| + C = -\frac{1}{2}\csc x \cot x + \frac{1}{2}\ln|\csc x - \cot x| + C\)
\(\displaystyle \int \csc^n{ax} \, dx = -\frac{\csc^{n-2}{ax} \cot{ax}}{a(n-1)} \,+\, \frac{n-2}{n-1}\int \csc^{n-2}{ax} \, dx \qquad \mbox{ (for }n \ne 1\mbox{)}\)
\(\displaystyle \int \frac{dx}{\csc{x} + 1} = x - \frac{2}{\cot{\frac{x}{2}}+1}+C\)
\(\displaystyle \int \frac{dx}{\csc{x} - 1} = - x + \frac{2}{\cot{\frac{x}{2}}-1}+C\)
Integrands involving only cotangent
\(\displaystyle \int\cot ax\,dx = \frac{1}{a}\ln|\sin ax|+C\)
\(\displaystyle \int \cot^2{x} \, dx = -\cot{x} - x +C\)
\(\displaystyle \int\cot^n ax\,dx = -\frac{1}{a(n-1)}\cot^{n-1} ax - \int\cot^{n-2} ax\,dx \qquad\mbox{(for }n\neq 1\mbox{)}\)
\(\displaystyle \int\frac{dx}{1 + \cot ax} = \int\frac{\tan ax\,dx}{\tan ax+1} = \frac{x}{2} - \frac{1}{2a}\ln|\sin ax + \cos ax|+C \)
\(\displaystyle \int\frac{dx}{1 - \cot ax} = \int\frac{\tan ax\,dx}{\tan ax-1} = \frac{x}{2} + \frac{1}{2a}\ln|\sin ax - \cos ax|+C \)
Integrands involving both sine and cosine
사인과 코사인의 유리 함수인 적분은 비오슈의 규칙(Charles Bioche)을 사용하여 평가될 수 있습니다.
\(\displaystyle \int\frac{dx}{\cos ax\pm\sin ax} = \frac{1}{a\sqrt{2}}\ln\left|\tan\left(\frac{ax}{2}\pm\frac{\pi}{8}\right)\right|+C\)
\(\displaystyle \int\frac{dx}{(\cos ax\pm\sin ax)^2} = \frac{1}{2a}\tan\left(ax\mp\frac{\pi}{4}\right)+C\)
\(\displaystyle \int\frac{dx}{(\cos x + \sin x)^n} = \frac{1}{n-1}\left(\frac{\sin x - \cos x}{(\cos x + \sin x)^{n - 1}} - 2(n - 2)\int\frac{dx}{(\cos x + \sin x)^{n-2}} \right)\)
\(\displaystyle \int\frac{\cos ax\,dx}{\cos ax + \sin ax} = \frac{x}{2} + \frac{1}{2a}\ln\left|\sin ax + \cos ax\right|+C\)
\(\displaystyle \int\frac{\cos ax\,dx}{\cos ax - \sin ax} = \frac{x}{2} - \frac{1}{2a}\ln\left|\sin ax - \cos ax\right|+C\)
\(\displaystyle \int\frac{\sin ax\,dx}{\cos ax + \sin ax} = \frac{x}{2} - \frac{1}{2a}\ln\left|\sin ax + \cos ax\right|+C\)
\(\displaystyle \int\frac{\sin ax\,dx}{\cos ax - \sin ax} = -\frac{x}{2} - \frac{1}{2a}\ln\left|\sin ax - \cos ax\right|+C\)
\(\displaystyle \int\frac{\cos ax\,dx}{(\sin ax)(1+\cos ax)} = -\frac{1}{4a}\tan^2\frac{ax}{2}+\frac{1}{2a}\ln\left|\tan\frac{ax}{2}\right|+C\)
\(\displaystyle \int\frac{\cos ax\,dx}{(\sin ax)(1-\cos ax)} = -\frac{1}{4a}\cot^2\frac{ax}{2}-\frac{1}{2a}\ln\left|\tan\frac{ax}{2}\right|+C\)
\(\displaystyle \int\frac{\sin ax\,dx}{(\cos ax)(1+\sin ax)} = \frac{1}{4a}\cot^2\left(\frac{ax}{2}+\frac{\pi}{4}\right)+\frac{1}{2a}\ln\left|\tan\left(\frac{ax}{2}+\frac{\pi}{4}\right)\right|+C\)
\(\displaystyle \int\frac{\sin ax\,dx}{(\cos ax)(1-\sin ax)} = \frac{1}{4a}\tan^2\left(\frac{ax}{2}+\frac{\pi}{4}\right)-\frac{1}{2a}\ln\left|\tan\left(\frac{ax}{2}+\frac{\pi}{4}\right)\right|+C\)
\(\displaystyle \int(\sin ax)(\cos ax)\,dx = \frac{1}{2a}\sin^2 ax +C\)
\(\displaystyle \int(\sin a_1x)(\cos a_2x)\,dx = -\frac{\cos((a_1-a_2)x)}{2(a_1-a_2)} -\frac{\cos((a_1+a_2)x)}{2(a_1+a_2)} +C\qquad\mbox{(for }|a_1|\neq|a_2|\mbox{)}\)
\(\displaystyle \int(\sin^n ax)(\cos ax)\,dx = \frac{1}{a(n+1)}\sin^{n+1} ax +C\qquad\mbox{(for }n\neq -1\mbox{)}\)
\(\displaystyle \int(\sin ax)(\cos^n ax)\,dx = -\frac{1}{a(n+1)}\cos^{n+1} ax +C\qquad\mbox{(for }n\neq -1\mbox{)}\)
\(\displaystyle \begin{align}
\int(\sin^n ax)(\cos^m ax)\,dx &= -\frac{(\sin^{n-1} ax)(\cos^{m+1} ax)}{a(n+m)}+\frac{n-1}{n+m}\int(\sin^{n-2} ax)(\cos^m ax)\,dx \qquad\mbox{(for }m,n>0\mbox{)} \\
&= \frac{(\sin^{n+1} ax)(\cos^{m-1} ax)}{a(n+m)} + \frac{m-1}{n+m}\int(\sin^n ax)(\cos^{m-2} ax)\,dx \qquad\mbox{(for }m,n>0\mbox{)}
\end{align}\)
\(\displaystyle \int\frac{dx}{(\sin ax)(\cos ax)} = \frac{1}{a}\ln\left|\tan ax\right|+C\)
\(\displaystyle \int\frac{dx}{(\sin ax)(\cos^n ax)} = \frac{1}{a(n-1)\cos^{n-1} ax}+\int\frac{dx}{(\sin ax)(\cos^{n-2} ax)} \qquad\mbox{(for }n\neq 1\mbox{)}\)
\(\displaystyle \int\frac{dx}{(\sin^n ax)(\cos ax)} = -\frac{1}{a(n-1)\sin^{n-1} ax}+\int\frac{dx}{(\sin^{n-2} ax)(\cos ax)} \qquad\mbox{(for }n\neq 1\mbox{)}\)
\(\displaystyle \int\frac{\sin ax\,dx}{\cos^n ax} = \frac{1}{a(n-1)\cos^{n-1} ax} +C\qquad\mbox{(for }n\neq 1\mbox{)}\)
\(\displaystyle \int\frac{\sin^2 ax\,dx}{\cos ax} = -\frac{1}{a}\sin ax+\frac{1}{a}\ln\left|\tan\left(\frac{\pi}{4}+\frac{ax}{2}\right)\right|+C\)
\(\displaystyle \int\frac{\sin^2 ax\,dx}{\cos^n ax} = \frac{\sin ax}{a(n-1)\cos^{n-1}ax}-\frac{1}{n-1}\int\frac{dx}{\cos^{n-2}ax} \qquad\mbox{(for }n\neq 1\mbox{)}\)
\(\displaystyle \begin{align}
\int \frac{\sin^2 x}{1 + \cos^2 x} \, dx &= \sqrt{2}\operatorname{arctangant}\left(\frac{\tan x}{\sqrt{2}}\right) - x \qquad\mbox{(for x in}] - \frac{\pi}{2} ; + \frac{\pi}{2} [\mbox{)} \\
&= \sqrt{2}\operatorname{arctangant}\left(\frac{\tan x}{\sqrt{2}}\right)-\operatorname{arctangant}\left(\tan x\right) \qquad\mbox{(this time x being any real number }\mbox{)}
\end{align}\)
\(\displaystyle \int\frac{\sin^n ax\,dx}{\cos ax} = -\frac{\sin^{n-1} ax}{a(n-1)} + \int\frac{\sin^{n-2} ax\,dx}{\cos ax} \qquad\mbox{(for }n\neq 1\mbox{)}\)
\(\displaystyle \int\frac{\sin^n ax\,dx}{\cos^m ax} = \begin{cases}
\frac{\sin^{n+1} ax}{a(m-1)\cos^{m-1} ax}-\frac{n-m+2}{m-1}\int\frac{\sin^n ax\,dx}{\cos^{m-2} ax} &\mbox{(for }m\neq 1\mbox{)} \\
\frac{\sin^{n-1} ax}{a(m-1)\cos^{m-1} ax}-\frac{n-1}{m-1}\int\frac{\sin^{n-2} ax\,dx}{\cos^{m-2} ax} &\mbox{(for }m\neq 1\mbox{)} \\
-\frac{\sin^{n-1} ax}{a(n-m)\cos^{m-1} ax}+\frac{n-1}{n-m}\int\frac{\sin^{n-2} ax\,dx}{\cos^m ax} &\mbox{(for }m\neq n\mbox{)}
\end{cases}\)
\(\displaystyle \int\frac{\cos ax\,dx}{\sin^n ax} = -\frac{1}{a(n-1)\sin^{n-1} ax} +C\qquad\mbox{(for }n\neq 1\mbox{)}\)
\(\displaystyle \int\frac{\cos^2 ax\,dx}{\sin ax} = \frac{1}{a}\left(\cos ax+\ln\left|\tan\frac{ax}{2}\right|\right) +C\)
\(\displaystyle \int\frac{\cos^2 ax\,dx}{\sin^n ax} = -\frac{1}{n-1}\left(\frac{\cos ax}{a\sin^{n-1} ax}+\int\frac{dx}{\sin^{n-2} ax}\right) \qquad\mbox{(for }n\neq 1\mbox{)}\)
\(\displaystyle \int\frac{\cos^n ax\,dx}{\sin^m ax} = \begin{cases}
-\frac{\cos^{n+1} ax}{a(m-1)\sin^{m-1} ax} - \frac{n-m+2}{m-1}\int\frac{\cos^n ax\,dx}{\sin^{m-2} ax} &\mbox{(for }m\neq 1\mbox{)} \\
-\frac{\cos^{n-1} ax}{a(m-1)\sin^{m-1} ax} - \frac{n-1}{m-1}\int\frac{\cos^{n-2} ax\,dx}{\sin^{m-2} ax} &\mbox{(for }m\neq 1\mbox{)} \\
\frac{\cos^{n-1} ax}{a(n-m)\sin^{m-1} ax} + \frac{n-1}{n-m}\int\frac{\cos^{n-2} ax\,dx}{\sin^m ax} &\mbox{(for }m\neq n\mbox{)}
\end{cases}\)
Integrands involving both sine and tangent
\(\displaystyle \int (\sin ax)(\tan ax)\,dx = \frac{1}{a}(\ln|\sec ax + \tan ax| - \sin ax)+C\)
\(\displaystyle \int\frac{\tan^n ax\,dx}{\sin^2 ax} = \frac{1}{a(n-1)}\tan^{n-1} (ax) +C\qquad\mbox{(for }n\neq 1\mbox{)}\)
Integrand involving both cosine and tangent
\(\displaystyle \int\frac{\tan^n ax\,dx}{\cos^2 ax} = \frac{1}{a(n+1)}\tan^{n+1} ax +C\qquad\mbox{(for }n\neq -1\mbox{)}\)
Integrand involving both sine and cotangent
\(\displaystyle \int\frac{\cot^n ax\,dx}{\sin^2 ax} = -\frac{1}{a(n+1)}\cot^{n+1} ax +C\qquad\mbox{(for }n\neq -1\mbox{)}\)
Integrand involving both cosine and cotangent
\(\displaystyle \int\frac{\cot^n ax\,dx}{\cos^2 ax} = \frac{1}{a(1-n)}\tan^{1-n} ax +C\qquad\mbox{(for }n\neq 1\mbox{)}\)
Integrand involving both secant and tangent
\(\displaystyle \int(\sec x)(\tan x)\,dx= \sec x + C\)
Integrand involving both cosecant and cotangent
\(\displaystyle \int(\csc x)(\cot x)\,dx= -\csc x + C\)
Integrals in a quarter period
\(\displaystyle \int_{{0}}^{{\frac{\pi}{2}}} \sin^n x \, dx = \int_{{0}}^{{\frac{\pi}{2}}} \cos^n x \, dx = \begin{cases}
\frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdots \frac{3}{4} \cdot \frac{1}{2} \cdot \frac{\pi}{2}, & \text{if } n\text{ is even} \\
\frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdots \frac{4}{5} \cdot \frac{2}{3}, & \text{if } n\text{ is odd and more than 1} \\
1, & \text{if } n=1
\end{cases}\)
Integrals with symmetric limits
\(\displaystyle \int_{{-c}}^{{c}}\sin{x}\,dx = 0 \)
\(\displaystyle \int_{{-c}}^{{c}}\cos {x}\,dx = 2\int_{{0}}^{{c}}\cos {x}\,dx = 2\int_{{-c}}^{{0}}\cos {x}\,dx = 2\sin {c} \)
\(\displaystyle \int_{{-c}}^{{c}}\tan {x}\,dx = 0 \)
\(\displaystyle \int_{-\frac{a}{2}}^{\frac{a}{2}} x^2\cos^2 {\frac{n\pi x}{a}}\,dx = \frac{a^3(n^2\pi^2-6)}{24n^2\pi^2} \qquad\mbox{(for }n=1,3,5...\mbox{)}\)
\(\displaystyle \int_{\frac{-a}{2}}^{\frac{a}{2}} x^2\sin^2 {\frac{n\pi x}{a}}\,dx = \frac{a^3(n^2\pi^2-6(-1)^n)}{24n^2\pi^2} = \frac{a^3}{24} (1-6\frac{(-1)^n}{n^2\pi^2}) \qquad\mbox{(for }n=1,2,3,...\mbox{)}\)
Integral over a full circle
\(\displaystyle \int_{{0}}^{{2 \pi}}\sin^{2m+1}{x}\cos^{n}{x}\,dx = 0 \! \qquad n,m \in \mathbb{Z}\)
\(\displaystyle \int_{{0}}^{{2 \pi}}\sin^{m}{x}\cos^{2n+1}{x}\,dx = 0 \! \qquad n,m \in \mathbb{Z}\)