不定积分表

基本积分公式

积分	结果
$\int k \thinspace dx$	$kx + C$
$\int x^n \thinspace dx$	$\dfrac{x^{n+1}}{n+1} + C \quad (n \neq -1)$
$\int \dfrac{1}{x} \thinspace dx$	$\ln \vert x \vert + C$
$\int e^x \thinspace dx$	$e^x + C$
$\int a^x \thinspace dx$	$\dfrac{a^x}{\ln a} + C \quad (a>0, a\neq 1)$

有理函数积分

积分	结果
$\int \dfrac{1}{a^2 + x^2} \thinspace dx$	$\dfrac{1}{a} \arctan \dfrac{x}{a} + C$
$\int \dfrac{1}{a^2 - x^2} \thinspace dx$	$\dfrac{1}{2a} \ln \left\vert \dfrac{a+x}{a-x} \right\vert + C$
$\int \dfrac{1}{x^2 - a^2} \thinspace dx$	$\dfrac{1}{2a} \ln \left\vert \dfrac{x-a}{x+a} \right\vert + C$
$\int \dfrac{1}{(x+a)(x+b)} \thinspace dx$	$\dfrac{1}{a-b} \ln \left\vert \dfrac{x+b}{x+a} \right\vert + C \quad (a \neq b)$
$\int \dfrac{x}{x^2 \pm a^2} \thinspace dx$	$\dfrac{1}{2} \ln \vert x^2 \pm a^2 \vert + C$

无理函数积分（含平方根）

积分	结果
$\int \dfrac{1}{\sqrt{a^2 - x^2}} \thinspace dx$	$\arcsin \dfrac{x}{a} + C$
$\int \dfrac{1}{\sqrt{x^2 \pm a^2}} \thinspace dx$	$\ln \left\vert x + \sqrt{x^2 \pm a^2} \right\vert + C$
$\int \dfrac{1}{x\sqrt{x^2 - a^2}} \thinspace dx$	$\dfrac{1}{a} \operatorname{arcsec} \left\vert \dfrac{x}{a} \right\vert + C$
$\int \sqrt{a^2 - x^2} \thinspace dx$	$\dfrac{x}{2} \sqrt{a^2 - x^2} + \dfrac{a^2}{2} \arcsin \dfrac{x}{a} + C$
$\int \sqrt{x^2 \pm a^2} \thinspace dx$	$\dfrac{x}{2} \sqrt{x^2 \pm a^2} \pm \dfrac{a^2}{2} \ln \left\vert x + \sqrt{x^2 \pm a^2} \right\vert + C$
$\int \dfrac{x}{\sqrt{a^2 \pm x^2}} \thinspace dx$	$\pm \sqrt{a^2 \pm x^2} + C$

三角函数积分

积分	结果
$\int \sin x \thinspace dx$	$-\cos x + C$
$\int \cos x \thinspace dx$	$\sin x + C$
$\int \tan x \thinspace dx$	$-\ln \vert \cos x \vert + C = \ln \vert \sec x \vert + C$
$\int \cot x \thinspace dx$	$\ln \vert \sin x \vert + C$
$\int \sec x \thinspace dx$	$\ln \vert \sec x + \tan x \vert + C$
$\int \csc x \thinspace dx$	$\ln \vert \csc x - \cot x \vert + C$
$\int \sec^2 x \thinspace dx$	$\tan x + C$
$\int \csc^2 x \thinspace dx$	$-\cot x + C$
$\int \sec x \tan x \thinspace dx$	$\sec x + C$
$\int \csc x \cot x \thinspace dx$	$-\csc x + C$
$\int \sin^2 x \thinspace dx$	$\dfrac{x}{2} - \dfrac{\sin 2x}{4} + C$
$\int \cos^2 x \thinspace dx$	$\dfrac{x}{2} + \dfrac{\sin 2x}{4} + C$
$\int \tan^2 x \thinspace dx$	$\tan x - x + C$
$\int \cot^2 x \thinspace dx$	$-\cot x - x + C$
$\int \sin^3 x \thinspace dx$	$\dfrac{\cos^3 x}{3} - \cos x + C$
$\int \cos^3 x \thinspace dx$	$\sin x - \dfrac{\sin^3 x}{3} + C$

反三角函数积分

积分	结果
$\int \arcsin x \thinspace dx$	$x \arcsin x + \sqrt{1-x^2} + C$
$\int \arccos x \thinspace dx$	$x \arccos x - \sqrt{1-x^2} + C$
$\int \arctan x \thinspace dx$	$x \arctan x - \dfrac{1}{2} \ln(1+x^2) + C$
$\int \operatorname{arccot} x \thinspace dx$	$x \operatorname{arccot} x + \dfrac{1}{2} \ln(1+x^2) + C$
$\int \operatorname{arcsec} x \thinspace dx$	$x \operatorname{arcsec} x - \ln \left\vert x + \sqrt{x^2-1} \right\vert + C$
$\int \operatorname{arccsc} x \thinspace dx$	$x \operatorname{arccsc} x + \ln \left\vert x + \sqrt{x^2-1} \right\vert + C$

指数与对数函数积分

积分	结果
$\int \ln x \thinspace dx$	$x \ln x - x + C$
$\int x e^{ax} \thinspace dx$	$\dfrac{e^{ax}}{a^2}(ax - 1) + C$
$\int x^n \ln x \thinspace dx$	$\dfrac{x^{n+1}}{n+1} \left( \ln x - \dfrac{1}{n+1} \right) + C \quad (n \neq -1)$
$\int \dfrac{\ln x}{x} \thinspace dx$	$\dfrac{1}{2} (\ln x)^2 + C$
$\int e^{ax} \sin bx \thinspace dx$	$\dfrac{e^{ax}}{a^2+b^2} (a \sin bx - b \cos bx) + C$
$\int e^{ax} \cos bx \thinspace dx$	$\dfrac{e^{ax}}{a^2+b^2} (a \cos bx + b \sin bx) + C$

双曲函数积分

积分	结果
$\int \sinh x \thinspace dx$	$\cosh x + C$
$\int \cosh x \thinspace dx$	$\sinh x + C$
$\int \tanh x \thinspace dx$	$\ln(\cosh x) + C$
$\int \coth x \thinspace dx$	$\ln \vert \sinh x \vert + C$
$\int \operatorname{sech} x \thinspace dx$	$\arctan(\sinh x) + C$
$\int \operatorname{csch} x \thinspace dx$	$\ln \left\vert \tanh \dfrac{x}{2} \right\vert + C$
$\int \operatorname{sech}^2 x \thinspace dx$	$\tanh x + C$
$\int \operatorname{csch}^2 x \thinspace dx$	$-\coth x + C$
$\int \operatorname{sech} x \tanh x \thinspace dx$	$-\operatorname{sech} x + C$
$\int \operatorname{csch} x \coth x \thinspace dx$	$-\operatorname{csch} x + C$