常用积分

下面所有出现的 a 均大于 0.

$$\begin{array}{rlrl}& \int k\mathrm{d}x& =& kx+C\\ & \int x^{\mu }\mathrm{d}x& =& \frac{x^{\mu +1}}{\mu +1}+C(\mu \ne -1)\\ & \int \frac{\mathrm{d}x}{x}& =& \ln |x|+C\\ & \int \frac{\mathrm{d}x}{1+x^2}& =& \arctan x+C\\ & \int \frac{\mathrm{d}x}{\sqrt{1-x^2}}& =& \arcsin x+C\\ & \int \cos x\mathrm{d}x& =& \sin x+C\\ & \int \sin x\mathrm{d}x& =& -\cos x+C\\ & \int {\sec }^{2}x\mathrm{d}x& =& \tan x+C\\ & \int {\csc }^{2}x\mathrm{d}x& =& -\cot x+C\\ & \int \sec x\tan x\mathrm{d}x& =& \sec x+C\\ & \int \csc x\cot x\mathrm{d}x& =& -\csc x+C\\ & \int {\text{e}}^x\mathrm{d}x& =& {\text{e}}^x+C\\ & \int a^x\mathrm{d}x& =& \frac{a^x}{\ln a}+C\\ & \int \sinh x\mathrm{d}x& =& \cosh x+C\\ & \int \cosh x\mathrm{d}x& =& \sinh x+C\\ & \int \tan x\mathrm{d}x& =& -\ln |\cos x|+C\\ & \int \cot x\mathrm{d}x& =& \ln |\sin x|+C\\ & \int \sec x\mathrm{d}x& =& \ln |\sec x+\tan x|+C\\ & \int \csc x\mathrm{d}x& =& -\ln |\csc x+\cot x|+C\\ & \int \frac{\mathrm{d}x}{a^2+x^2}& =& \frac{1}{a}\arctan \frac{x}{a}+C\\ & \int \frac{\mathrm{d}x}{x^2-a^2}& =& \frac{1}{2a}\ln |\frac{x-a}{x+a}|+C\\ & \int \frac{\mathrm{d}x}{a^2-x^2}& =& \frac{1}{2a}\ln |\frac{a+x}{a-x}|+C\\ & \int \frac{\mathrm{d}x}{\sqrt{a^2-x^2}}& =& \arcsin \frac{x}{a}+C\\ & \int \frac{\mathrm{d}x}{\sqrt{x^2+a^2}}& =& \ln (x+\sqrt{x^2+a^2})+C\\ & \int \frac{\mathrm{d}x}{\sqrt{x^2-a^2}}& =& \ln |x+\sqrt{x^2-a^2}|+C\end{array}$$