微积分/积分表

规则

• $\int c\cdot f(x)\mathrm {d} x=c\cdot \int f(x)\mathrm {d} x$
• $\int {\big (}f(x)\pm g(x){\big )}\mathrm {d} x=\int f(x)\mathrm {d} x\pm \int g(x)\mathrm {d} x$
• $\int f(g(x))g'(x)\mathrm {d} x=\int f(u)\mathrm {d} u=F(u)+C=F(g(x))+C$ 其中 $F'=f$
• $\int u\,dv=uv-\int v\,du$

幂

• $\int \mathrm {d} x=x+C$
• $\int a\,\mathrm {d} x=ax+C$
• $\int x^{n}\mathrm {d} x={\frac {x^{n+1}}{n+1}}+C\qquad ({\text{for }}n\neq -1)$
• $\int {\frac {\mathrm {d} x}{x}}=\ln |x|+C$
• $\int {\frac {\mathrm {d} x}{ax+b}}={\frac {\ln |ax+b|}{a}}+C\qquad ({\text{for }}a\neq 0)$

三角函数

基本三角函数

• $\int \sin(x)\mathrm {d} x=-\cos(x)+C$
• $\int \cos(x)\mathrm {d} x=\sin(x)+C$
• $\int \tan(x)\mathrm {d} x=\ln |\sec(x)|+C$
• $\int \sin ^{2}(x)\mathrm {d} x=\int {\frac {1-\cos(2x)}{2}}\mathrm {d} x={\frac {x}{2}}-{\frac {\sin(2x)}{4}}+C$
• $\int \cos ^{2}(x)\mathrm {d} x=\int {\frac {1+\cos(2x)}{2}}\mathrm {d} x={\frac {x}{2}}+{\frac {\sin(2x)}{4}}+C$
• $\int \tan ^{2}(x)\mathrm {d} x=\tan(x)-x+C$

倒数三角函数

• $\int \sec(x)\mathrm {d} x=\ln {\Big |}\sec(x)+\tan(x){\Big |}+C=\ln \left|\tan \left({\frac {x}{2}}+{\frac {\pi }{4}}\right)\right|+C=2\mathrm {artanh} \left(\tan \left({\frac {x}{2}}\right)\right)+C$
• $\int \csc(x)\mathrm {d} x=-\ln {\Big |}\csc(x)+\cot(x){\Big |}+C=\ln \left|\tan \left({\frac {x}{2}}\right)\right|+C$
• $\int \cot(x)\mathrm {d} x=\ln |\sin(x)|+C$

• $\int \sec ^{2}(ax)\mathrm {d} x={\frac {\tan(ax)}{a}}+C$
• $\int \csc ^{2}(ax)\mathrm {d} x=-{\frac {\cot(ax)}{a}}+C$
• $\int \cot ^{2}(ax)\mathrm {d} x=-x-{\frac {\cot(ax)}{a}}+C$
• $\int \sec(x)\tan(x)\mathrm {d} x=\sec(x)+C$
• $\int \sec(x)\csc(x)\mathrm {d} x=\ln |\tan(x)|+C$

降阶公式

• $\int \sin ^{n}(x)\mathrm {d} x=-{\frac {\sin ^{n-1}(x)\cos(x)}{n}}+{\frac {n-1}{n}}\int \sin ^{n-2}(x)\mathrm {d} x+C\qquad ({\text{for }}n>0)$
• $\int \cos ^{n}(x)\mathrm {d} x=-{\frac {\cos ^{n-1}(x)\sin(x)}{n}}+{\frac {n-1}{n}}\int \cos ^{n-2}(x)\mathrm {d} x+C\qquad ({\text{for }}n>0)$
• $\int \tan ^{n}(x)\mathrm {d} x={\frac {\tan ^{n-1}(x)}{(n-1)}}-\int \tan ^{n-2}(x)\mathrm {d} x+C\qquad ({\text{for }}n\neq 1)$
• $\int \sec ^{n}(x)\mathrm {d} x={\frac {\sec ^{n-1}(x)\sin(x)}{n-1}}+{\frac {n-2}{n-1}}\int \sec ^{n-2}(x)\mathrm {d} x+C\qquad ({\text{for }}n\neq 1)$
• $\int \csc ^{n}(x)\mathrm {d} x=-{\frac {\csc ^{n-1}(x)\cos(x)}{n-1}}+{\frac {n-2}{n-1}}\int \csc ^{n-2}(x)\mathrm {d} x+C\qquad ({\text{for }}n\neq 1)$
• $a^{2}\int x^{n}\sin(ax)\mathrm {d} x=nx^{n-1}\sin(ax)-ax^{n}\cos(ax)-n(n-1)\int x^{n-2}\sin(ax)\mathrm {d} x$
• $a^{2}\int x^{n}\cos(ax)\mathrm {d} x=ax^{n}\sin(ax)+nx^{n-1}\cos(ax)-n(n-1)\int x^{n-2}\cos(ax)\mathrm {d} x$

明确形式

• $\int \sin ^{n}(x)\mathrm {d} x=-\cos(x)_{2}F_{1}\left({\frac {1}{2}},{\frac {1-n}{2}};{\frac {3}{2}};\cos ^{2}(x)\right)+C$
• $\int \cos ^{n}(x)\mathrm {d} x=-{\frac {1}{n+1}}\mathrm {sgn} (\sin(x))\cos ^{n+1}(x)_{2}F_{1}\left({\frac {1}{2}},{\frac {n+1}{2}};{\frac {n+3}{2}};\cos ^{2}(x)\right)+C\qquad ({\text{for }}n\neq -1)$
• $\int \tan ^{n}(x)\mathrm {d} x={\frac {1}{n+1}}\tan ^{n+1}(x)_{2}F_{1}\left(1,{\frac {n+1}{2}};{\frac {n+3}{2}};-\tan ^{2}(x)\right)+C\qquad ({\text{for }}n\neq -1)$
• $\int \csc ^{n}(x)\mathrm {d} x=-\cos(x)_{2}F_{1}\left({\frac {1}{2}},{\frac {n+1}{2}};{\frac {3}{2}};\cos ^{2}(x)\right)+C$
• $\int \sec ^{n}(x)\mathrm {d} x=\sin(x)_{2}F_{1}\left({\frac {1}{2}},{\frac {n+1}{2}};{\frac {3}{2}};\sin ^{2}(x)\right)+C$
• $\int \cot ^{n}(x)\mathrm {d} x=-{\frac {1}{n+1}}\cot ^{n+1}(x)_{2}F_{1}\left(1,{\frac {n+1}{2}};{\frac {n+3}{2}};-\cot ^{2}(x)\right)+C\qquad ({\text{for }}n\neq -1)$

其中 ${}_{2}F_{1}$ 是超几何函数， $\mathrm {sgn}$ 是符号函数。

反三角函数

• $\int {\frac {\mathrm {d} x}{\sqrt {1-x^{2}}}}=\arcsin(x)+C$
• $\int {\frac {\mathrm {d} x}{\sqrt {a^{2}-x^{2}}}}=\arcsin \left({\tfrac {x}{a}}\right)+C\qquad ({\text{for }}a\neq 0)$
• $\int {\frac {\mathrm {d} x}{1+x^{2}}}=\arctan(x)+C$
• $\int {\frac {\mathrm {d} x}{a^{2}+x^{2}}}={\frac {\arctan \left({\tfrac {x}{a}}\right)}{a}}+C\qquad ({\text{for }}a\neq 0)$

指数函数和对数函数

• $\int e^{x}\mathrm {d} x=e^{x}+C$
• $\int e^{ax}\mathrm {d} x={\frac {e^{ax}}{a}}+C\qquad ({\text{for }}a\neq 0)$
• $\int a^{x}\mathrm {d} x={\frac {a^{x}}{\ln(a)}}+C\qquad ({\text{for }}a>0,a\neq 1)$
• $\int \ln(x)\mathrm {d} x=x\ln(x)-x+C$
• $\int e^{x}\sin(x)\mathrm {d} x={\frac {e^{x}}{2}}(\sin(x)-\cos(x))+C$
• $\int e^{x}\cos(x)\mathrm {d} x={\frac {e^{x}}{2}}(\sin(x)+\cos(x))+C$

降阶公式

• $\int x^{n}e^{ax}\mathrm {d} x={\frac {1}{a}}x^{n}e^{ax}-{\frac {n}{a}}\int x^{n-1}e^{ax}\mathrm {d} x$

反三角函数

• $\int \arcsin(x)\mathrm {d} x=x\arcsin(x)+{\sqrt {1-x^{2}}}+C$
• $\int \arccos(x)\mathrm {d} x=x\arccos(x)-{\sqrt {1-x^{2}}}+C$
• $\int \arctan(x)\mathrm {d} x=x\arctan(x)-{\frac {1}{2}}\ln |1+x^{2}|+C$
• $\int \operatorname {arccsc}(x)\mathrm {d} x=x\operatorname {arccsc}(x)+\ln \left|x+x{\sqrt {1-{\frac {1}{x^{2}}}}}\right|+C$
• $\int \operatorname {arcsec}(x)\mathrm {d} x=x\operatorname {arcsec}(x)-\ln \left|x+x{\sqrt {1-{\frac {1}{x^{2}}}}}\right|+C$
• $\int \operatorname {arccot}(x)\mathrm {d} x=x\operatorname {arccot}(x)+{\frac {1}{2}}\ln |1+x^{2}|+C$

双曲函数

• $\int \sinh(x)\mathrm {d} x=-i\int \sin(ix)\mathrm {d} x=\cos(ix)+C=\cosh(x)+C$
• $\int \cosh(x)\mathrm {d} x=\int \cos(ix)\mathrm {d} x=-i\sin(ix)+C=\sinh(x)+C$
• $\int \tanh(x)\mathrm {d} x=-i\int \tan(ix)\mathrm {d} x=\log \left|\cos(ix)\right|+C=\log \left|\cosh(x)\right|+C$

倒数

• $\int \mathrm {csch} (x)\mathrm {d} x=i\int \csc(ix)\mathrm {d} x=\log \left|-i\tan \left({\frac {ix}{2}}\right)\right|+C=\log \left|\tanh \left({\frac {x}{2}}\right)\right|+C$
• $\int \mathrm {sech} (x)\mathrm {d} x=\int \sec(ix)\mathrm {d} x=2\mathrm {artanh} \left(-i\tan \left({\frac {x}{2}}i\right)\right)+C=2\arctan \left(\tanh \left({\frac {x}{2}}\right)\right)+C$
• $\int \mathrm {coth} (x)\mathrm {d} x=i\int \cot(ix)\mathrm {d} x=\log \left|-i\sin(ix)\right|+C=\log \left|\sinh(x)\right|+C$

反函数

• $\int \mathrm {arsinh} (x)\mathrm {d} x=x\mathrm {arsinh} (x)-{\sqrt {x^{2}+1}}+C$
• $\int \mathrm {arcosh} (x)\mathrm {d} x=x\mathrm {arcosh} (x)-{\sqrt {x^{2}-1}}+C$
• $\int \mathrm {artanh} (x)\mathrm {d} x=x\mathrm {artanh} (x)+{\frac {1}{2}}\ln(1-x^{2})+C$
• $\int \mathrm {arcsch} (x)\mathrm {d} x=x\mathrm {arcsch} (x)+|\mathrm {arsinh} (x)|+C$
• $\int \mathrm {arsech} (x)\mathrm {d} x=x\mathrm {arsech} (x)+\arcsin(x)+C$
• $\int \mathrm {artanh} (x)\mathrm {d} x=x\mathrm {arcoth} (x)+{\frac {1}{2}}\ln(x^{2}-1)+C$

其他

• $\int |f(x)|\mathrm {d} x=\mathrm {sgn} (f(x))\int f(x)\mathrm {d} x$，其中 $\mathrm {sgn}$ 是符号函数。

定积分

• $\int _{[0,1]^{n}}{\frac {\prod _{i=1}^{n}\mathrm {d} x_{i}}{1-\prod _{i=1}^{n}x_{i}}}=\zeta (n){\text{ for all integers }}n>1$，其中 $\zeta$ 是黎曼ζ函数。
• $\int _{-\infty }^{\infty }e^{-x^{2}}\mathrm {d} x={\sqrt {\pi }}$
• $\int _{0}^{1}t^{u-1}(1-t)^{v-1}\mathrm {d} t=\beta (u,v)={\frac {\Gamma (u)\Gamma (v)}{\Gamma (u+v)}}$，其中 $\Gamma$ 是伽马函数。
• $\int _{0}^{\infty }t^{s-1}e^{-t}\mathrm {d} t=\Gamma (s)$
• $\int _{0}^{2\pi }e^{u\cos \theta }\mathrm {d} \theta =2\pi I_{0}(u)$，其中 $I_{0}$ 是第一类修正贝塞尔函数。
• $\int _{0}^{\infty }{\frac {\sin(x)}{x}}\mathrm {d} x={\frac {\pi }{2}}$