积分公式总结

基本积分公式表

(1) $\int k \mathrm{~d} x=k x+C(k$ 是常数 $)$,

(2) $\int x^\mu \mathrm{d} x=\frac{x^{\mu+1}}{\mu+1}+C(\mu \neq-1)$,

(3) $\int \frac{\mathrm{d} x}{x}=\ln |x|+C$,

(4) $\int \frac{\mathrm{d} x}{1+x^2}=\arctan x+C$,

(5) $\int \frac{\mathrm{d} x}{\sqrt{1-x^2}}=\arcsin x+C$,

(6) $\int \cos x \mathrm{~d} x=\sin x+C$,

(7) $\int \sin x \mathrm{~d} x=-\cos x+C$,

(8) $\int \frac{\mathrm{d} x}{\cos ^2 x}=\int \sec ^2 x \mathrm{~d} x=\tan x+C$,

(9) $\int \frac{\mathrm{d} x}{\sin ^2 x}=\int \csc ^2 x \mathrm{~d} x=-\cot x+C$,

(10) $\int \sec x \tan x \mathrm{~d} x=\sec x+C$,

(11) $\int \csc x \cot x \mathrm{~d} x=-\csc x+C$,

(12) $\int \mathrm{e}^x \mathrm{~d} x=\mathrm{e}^x+C$,

(13) $\int a^x \mathrm{~d} x=\frac{a^x}{\ln a}+C$.

(11) $\int \csc x \cot x \mathrm{~d} x=-\csc x+C$,

(12) $\int \mathrm{e}^x \mathrm{~d} x=\mathrm{e}^x+C$,

(13) $\int a^x \mathrm{~d} x=\frac{a^x}{\ln a}+C$.

(14) $\int \operatorname{sh} x \mathrm{~d} x=\operatorname{ch} x+C$,

(15) $\int \operatorname{ch} x \mathrm{~d} x=\operatorname{sh} x+C$.

(16) $\int \tan x \mathrm{~d} x=-\ln |\cos x|+C$,

(11) $\int \cot x \mathrm{~d} x=\ln |\sin x|+C$,

(18) $\int \sec x \mathrm{~d} x=\ln |\sec x+\tan x|+C$,

(19) $\int \csc x \mathrm{~d} x=\ln |\csc x-\cot x|+C$,

(2) $\int \frac{\mathrm{d} x}{a^2+x^2}=\frac{1}{a} \arctan \frac{x}{a}+C$,

(21) $\int \frac{\mathrm{d} x}{x^2-a^2}=\frac{1}{2 a} \ln \left|\frac{x-a}{x+a}\right|+C$,

(16) $\int \tan x \mathrm{~d} x=-\ln |\cos x|+C$,

(11) $\int \cot x \mathrm{~d} x=\ln |\sin x|+C$,

(18) $\int \sec x \mathrm{~d} x=\ln |\sec x+\tan x|+C$,

(19) $\int \csc x \mathrm{~d} x=\ln |\csc x-\cot x|+C$,

(20) $\int \frac{\mathrm{d} x}{a^2+x^2}=\frac{1}{a} \arctan \frac{x}{a}+C$,

(21) $\int \frac{\mathrm{d} x}{x^2-a^2}=\frac{1}{2 a} \ln \left|\frac{x-a}{x+a}\right|+C$,

(22)$\int \frac{\mathrm{d} x}{\sqrt{a^2-x^2}}=\arcsin \frac{x}{a}+C$,

(23)$\int \frac{\mathrm{d} x}{\sqrt{x^2+a^2}}=\ln \left(x+\sqrt{x^2+a^2}\right)+C$,

(24) $\int \frac{\mathrm{d} x}{\sqrt{x^2-a^2}}=\ln \left|x+\sqrt{x^2-a^2}\right|+C$.

$\int u^{\prime} v d x=u v-\int u v^{\prime} d x$
$\int u d v=u v-\int v d u$