基本积分表

$\text{Newton-Leibniz formula} \\ \int_a^bf(x)\mathrm dx =\left[F(x)\right]^b_a=F(b)-F(a)$ $\int k\text dx=kx+C\\ \int x^\mu\text dx=\frac{x^{\mu+1}}{\mu+1}+C(\mu\ne-1)\\ \int \frac{\text dx}{x}=\ln|x|+C\\ \int \frac{\text dx}{1+x^2}=\arctan x+C\\ \int \frac{\text dx}{\sqrt{1-x^2}}= \arcsin x+C$ $\int \cos x\text dx=\sin x +C\\ \int \sin x\text dx=-\cos x+C\\ \\\int \frac{\text dx}{\cos^2x}=\int\sec^2x=\tan x+C\\ \int \frac{\text dx}{\sin^2x}=\int\csc^2x=-\cot x+C\\ \\ \int \sec x\tan x\text dx=\sec x+C$ $\int \csc x \cot x\text dx=-\csc x+C\\ \int e^x=e^x+C\\\int a^x\text dx=\frac{a^x}{\ln a}+C\\$

拓展公式

$\int \frac{\text dx}{a^2+x^2}=\frac1a \arctan \frac xa+C\\ \int \frac{\text dx}{x^2-a^2}=\frac1{2a}\ln\left|\frac{x-a}{x+a}\right|+C\\ \int \frac{\text dx}{\sqrt{a^2-x^2}}= \arcsin \frac xa +C\\ \int \frac{\text dx}{\sqrt{a^2+x^2}}=\ln (x+\sqrt{x^2+a^2})+C\\ \int \frac{\text dx}{\sqrt{x^2-a^2}}=\ln |x+\sqrt{x^2-a^2}|+C\\$ $\int\tan x\text{d}x=-\ln |\cos x|+C\\ \int\cot x\text{d}x=\ln |\sin x|+C\\ \int\sec x\text{d}x=\ln |\sec x+\tan x|+C\\ \int\csc x\text{d}x=\ln |\csc x-\cot x|+C\\ \int \sinh x\text{d}x=\cosh x +C\\ \int \cosh x\text{d}x=\sinh x +C\\ \int x\cos x=x\sin x +\cos x +C$

常见凑微分公式

$a\text{d}x=\text{d}(ax)\\ 2x\text{d}x=\text{d}x^2\\ e^x\text{d}x=\text{d}e^x\\ \frac {e^x}{2\sqrt{1+e^x}}\text{d}x=\text{d}\sqrt{1+e^x}\\ e^x+e^{-x}\text{d}x=\text{d}(e^x-e^{-x})\\ \frac {x}{\sqrt{1+x^2}}\text{d}x=\text{d}\sqrt{1+x^2}$