积分基本公式

$$(1)\int x^{\alpha }dx=\frac{x^{\alpha }}{\alpha +1}+C(\alpha \ne -1)$$

特别的，

$$\int 0dx=C;\int kdx=kx+C(k\mathrm{为}\mathrm{常}\mathrm{数})$$
$$(2)\int \frac{1}{x}dx=\ln |x|+C$$$$(3)\int a^{x}dx=\frac{a^x}{\ln a}+C(a>0,a\ne -1)$$$$(4)\int \cos xdx=\sin x+C$$$$(5)\int \sin xdx=-\cos x+C$$$$(6)\int {\sec }^{2}xdx=\tan x+C$$$$(7)\int {\csc }^{2}xdx=-\cot x+C$$$$(8)\int \sec x\tan xdx=\sec x+C$$$$(9)\int \csc x\cot xdx=-\csc x+C$$$$(10)\int \frac{1}{\sqrt{1-x^2}}dx=\arcsin x+C$$$$(11)\int \frac{1}{1+x^2}dx=\arctan x+C$$$$(12)\int \tan xdx=-\ln |\cos x|+C$$$$(13)\int \cot xdx=\ln |\sin x|+C$$$$(14)\int \sec xdx=\ln |\sec x+\tan x|+C$$$$(15)\int \csc xdx=\ln |\csc x-\cot x|+C$$$$(16)\int \frac{1}{x^2-a^2}dx=\frac{1}{2a}\ln |\frac{x-a}{x+a}|+C$$$$(17)\int \frac{1}{a^2+x^2}dx=\frac{1}{a}\arctan \frac{x}{a}+C$$$$(18)\int \frac{1}{\sqrt{a^2-x^2}}dx=\arcsin \frac{x}{a}+C$$$$(19)\int \frac{1}{\sqrt{x^2\pm a^2}}dx=\ln |x+\sqrt{x^2\pm a^2}|+C$$$$(20)\int \sqrt{x^2\pm a^2}dx=\frac{x}{2}\sqrt{x^2\pm a^2}\pm \frac{a}{2}\ln |x+\sqrt{x^2\pm a^2}|+C$$$$(21)I_n=\int \frac{1}{(x^2+a^2{)}^n}dx(n\in N_{+}),$$$$I_1=\frac{1}{a}\arctan \frac{x}{a}+C,$$$$I_n=\frac{1}{2a^2(n-1)}[\frac{x}{(x^2+a^2{)}^{n-1}}+(2n-3)I_{n-1}](n>1)$$