# 积分表

$\int_M \mathrm{d}\omega = \oint_{\partial M} \omega$

## 含有$a+bx$的积分

$\int\frac{1}{a+bx}dx=\frac{1}{b}\ln \left | a+bx\right|+C$
$\int\frac{x}{a+bx}dx=\frac{1}{b^2}(bx-a\ln\left ( a+bx \right )) +C$
$\int\frac{x^2}{a+bx}dx=\frac{1}{2b^3} ((a+bx)^2-4a(a+bx)+2a^2 \ln\left ( a+bx\right ) )+C$
$\int\frac{1}{x(a+bx)}dx = \frac{-1}{a}\ln\left ( \frac{a+bx}{x}\right )+C$
$\int\frac{1}{x^2(a+bx)}dx=\frac{b}{a^2}\ln\left (\frac{a+bx}{x}\right )-\frac{1}{ax}+C$

## 含有$\sqrt{a+bx}$的积分

$\int x\sqrt{a+bx}dx=\frac{2}{15b^2}(3bx-2a)(a+bx)^{\frac{3}{2}}+C$
$\int x^2\sqrt{a+bx}dx=\frac{2}{105b^3}(15b^2x^2-12abx+8a^2)(a+bx)^{\frac{3}{2}}+C$
$\int x^n\sqrt{a+bx}dx=\frac{2}{b(2n+3)}x^n(a+bx)^{\frac{3}{2}} -\frac{2na}{b(2n+3)}\int x^{n-1}\sqrt{a+bx}dx$
$\int\frac{\sqrt{a+bx}}{x}dx=2\sqrt{a+bx}+a\int\frac{1}{x\sqrt{a+bx}}dx$
$\int\frac{\sqrt{a+bx}}{x^n}dx=\frac{-1}{a(n-1)}\frac{(a+bx)^{\frac{3}{2}}}{x^{n-1}} -\frac{(2n-5)b}{2a(n-1)}\int\frac{\sqrt{a+bx}}{x^{n-1}}dx,n\neq 1$
$\int\frac{1}{x\sqrt{a+bx}}dx=\frac{1}{\sqrt{a}}\ln\left (\frac{\sqrt{a+bx} -\sqrt{a}}{\sqrt{a+bx}+\sqrt{a}}\right )+C,a>0$
$=\frac{2}{\sqrt{-a}}\arctan\sqrt\frac{a+bx}{-a} +C,a<0$
$\int\frac{1}{x^n\sqrt{a+bx}}dx=\frac{-1}{a(n-1)}\frac{\sqrt{a+bx}}{x^{n-1}} -\frac{(2n-3)b}{2a(n-1)}\int\frac{1}{x^{n-1}}\sqrt{a+bx}dx,n\neq 1$

## 含有$x^2\pm\alpha^2$的积分

$\int\frac{1}{x^2+\alpha^2}\mbox{d}x=\frac{\arctan\dfrac{x}{\alpha}}{\alpha}+C$
$\int\frac{1}{\pm x^2\mp\alpha^2}\mbox{d}x = \frac{\ln\left(\dfrac{x\mp\alpha}{\pm x+\alpha}\right)}{2\alpha}+C$

## 含有 ${ax^2+b}$的积分

$\int\frac{1}{ax^2+b}\mbox{d}x=\frac{1}{\sqrt{ab}} \arctan\frac{\sqrt{a}x}{\sqrt{b}}+C$

## 含有 $ax^2+bx+c\qquad(a>0)$的积分

$\int ax^2+bx+c\mbox{d}x=\frac{ax^3}{3}+\frac{bx^2}{2}+cx+C$

## 含有 $\sqrt{a^2+x^2}\qquad(a>0)$的积分

$\int\sqrt{a^2+x^2}dx=\frac{1}{2}x\sqrt{a^2+x^2}+\frac{1}{2}a^2\ln\left (x+\sqrt{a^2+x^2}\right )+C$
$\int x^2\sqrt{a^2+x^2}dx=\frac{1}{8}x(a^2+2x^2)\sqrt{a^2+x^2}-\frac{1}{8}a^4\ln\left (x+\sqrt{a^2+x^2}\right )+C$
$\int \frac{\sqrt{a^2+x^2}}{x}dx = \sqrt{a^2+x^2} - a\ln \left ( \frac{a+\sqrt{a^2+x^2}}{x} \right ) +C$
$\int \frac{\sqrt{a^2+x^2}}{x^2}dx = \ln\left ( x+\sqrt{a^2+x^2}\right ) - \frac{\sqrt{a^2+x^2}}{x} +C$
$\int \frac{1}{\sqrt{a^2+x^2}}dx=\ln \left ( x+\sqrt{a^2+x^2} \right ) +C$
$\int\frac{x^2}{\sqrt{a^2+x^2}}dx=\frac{1}{2}x\sqrt{a^2+x^2} -\frac{1}{2}a^2\ln\left (\sqrt{a^2+x^2}+x \right )+C$
$\int\frac{1}{x\sqrt{a^2+x^2}}dx=\frac{1}{a}\ln\left (\frac{x}{a+\sqrt{a^2+x^2}}\right )+C$
$\int\frac{1}{x^2\sqrt{a^2+x^2}}dx=-\frac{\sqrt{a^2+x^2}}{a^2x}+C$

## 含有$\sqrt{x^2-a^2}\qquad{(x^2>a^2)}$的积分

$\int \frac{1}{\sqrt{x^2-a^2}}dx=\ln\left ( x+\sqrt{x^2-a^2} \right ) +C$

## 含有$\sqrt{a^2-x^2} \qquad(a^2>x^2)$的积分

$\int \frac{1}{\sqrt{a^2-x^2}}dx= \arcsin \frac{x}{a} +C = - \arccos \frac{x}{a} +C$
$\int\sqrt{a^2-x^2}dx=\frac{1}{2}x\sqrt{a^2-x^2} +\frac{a^2}{2}\arcsin\frac{x}{a}+C$
$\int x^2\sqrt{a^2-x^2}dx=\frac{1}{8}x(2x^2-a^2)\sqrt{a^2-x^2} +\frac{1}{8}a^4\arcsin\frac{x}{a}+C$
$\int\frac{\sqrt{a^2-x^2}}{x}dx=\sqrt{a^2-x^2} -a\ln\left (\frac{a+\sqrt{a^2-x^2}}{x}\right )+C$
$\int\frac{\sqrt{a^2-x^2}}{x^2}dx=-\frac{\sqrt{a^2-x^2}}{x} -\arcsin\frac{x}{a}+C$
$\int\frac{1}{u\sqrt{a^2-x^2}}dx=-\frac{1}{a}\ln\left (\frac{a+\sqrt{a^2-x^2}}{x}\right )+C$
$\int\frac{x^2}{\sqrt{a^2-x^2}}dx=-\frac{1}{2}x\sqrt{a^2-x^2}+\frac{1}{2}a^2\arcsin\frac{x}{a}+C$
$\int\frac{1}{x^2\sqrt{a^2-x^2}}dx=-\frac{\sqrt{a^2-x^2}}{a^2x}+C$

## 含有$\sqrt{|a|x^2+bx+c}\qquad(a\ne0)$的积分

$\int\frac{dx}{R} = \frac{1}{\sqrt{a}}\ln\left(2\sqrt{a}R+2ax+b\right)\qquad(\mbox{for }a>0)$
$\int\frac{dx}{R} = \frac{1}{\sqrt{a}}\,\operatorname{arsinh}\frac{2ax+b}{\sqrt{4ac-b^2}} \qquad \mbox{(for }a>0\mbox{, }4ac-b^2>0\mbox{)}$
$\int\frac{dx}{R} = \frac{1}{\sqrt{a}}\ln|2ax+b| \quad \mbox{(for }a>0\mbox{, }4ac-b^2=0\mbox{)}$
$\int\frac{dx}{R} = -\frac{1}{\sqrt{-a}}\arcsin\frac{2ax+b}{\sqrt{b^2-4ac}} \qquad \mbox{(for }a<0\mbox{, }4ac-b^2<0\mbox{, }\left(2ax+b\right)<\sqrt{b^2-4ac}\mbox{)}$
$\int\frac{dx}{R^3} = \frac{4ax+2b}{(4ac-b^2)R}$
$\int\frac{dx}{R^5} = \frac{4ax+2b}{3(4ac-b^2)R}\left(\frac{1}{R^2}+\frac{8a}{4ac-b^2}\right)$
$\int\frac{dx}{R^{2n+1}} = \frac{2}{(2n-1)(4ac-b^2)}\left(\frac{2ax+b}{R^{2n-1}}+4a(n-1)\int\frac{dx}{R^{2n-1}}\right)$
$\int\frac{x}{R}\;dx = \frac{R}{a}-\frac{b}{2a}\int\frac{dx}{R}$
$\int\frac{x}{R^3}\;dx = -\frac{2bx+4c}{(4ac-b^2)R}$
$\int\frac{x}{R^{2n+1}}\;dx = -\frac{1}{(2n-1)aR^{2n-1}}-\frac{b}{2a}\int\frac{dx}{R^{2n+1}}$
$\int\frac{dx}{xR}=-\frac{1}{\sqrt{c}}\ln\left(\frac{2\sqrt{c}R+bx+2c}{x}\right)$
$\int\frac{dx}{xR}=-\frac{1}{\sqrt{c}}\operatorname{arsinh}\left(\frac{bx+2c}{|x|\sqrt{4ac-b^2}}\right)$

## 含有三角函数的积分

$\int\cos x\mbox{d}x=\sin x+C$
$\int-\sin x\mbox{d}x=\cos x+C$
$\int\sec^2x\mbox{d}x=\tan x+C$
$\int-\csc^2x\mbox{d}x=\cot x+C$
$\int\sec x\tan x\mbox{d}x=\sec x+C$
$\int-\csc x\cot x\mbox{d}x=\csc x+C$

$\int\tan x\mbox{d}x= - \ln(\cos x)+C$
$\int\cot x\mbox{d}x=\ln(\sin x)+C$
$\int\sec x\mbox{d}x=\ln(\sec x+\tan x)+C$
$\int\csc x\mbox{d}x=-\ln(\csc x+\cot x)+C=\ln\left({\tan x-\sin x\over\sin x\tan x}\right)+C$

$\int \sin ^n x dx = - \frac{1}{n} \sin ^{n-1} x \cos x + \frac{n-1}{n} \int \sin ^{n-2} x dx +C \quad \forall n \ge 2$
$\int \sin ^2 x dx = \frac{x}{2}-\frac{\sin{2x}}{4} +C$

$\int \cos ^n x dx = \frac{1}{n} \cos ^{n-1} x \sin x + \frac{n-1}{n} \int \cos ^{n-2} x dx +C \quad \forall n \ge 2$
$\int \cos ^2 x dx = \frac{x}{2}+\frac{\sin{2x}}{4} +C$

$\int \tan ^n x dx = \frac{1}{n-1} \tan ^{n-1} x - \int \tan ^{n-2} x dx +C \quad \forall n \ge 2$
$\int \tan ^2 x dx = \tan x - x +C$

$\int \cot ^n x dx = \frac{1}{n-1} \cot ^{n-1} x - \int \cot ^{n-2} x dx +C \quad \forall n \ge 2$
$\int \cot ^2 x dx = - \cot x - x +C$

$\int \sec ^n x dx = \frac{1}{n-1} \sec ^{n-2} x \tan x + \frac{n-2}{n-1} \int \sec ^{n-2} x dx +C \quad \forall n \ge 2$

$\int \csc ^n x dx = - \frac{1}{n-1} \csc ^{n-2} x \cot x + \frac{n-2}{n-1} \int \csc ^{n-2} x dx +C \quad \forall n \ge 2$

## 含有反三角函数的积分

$\int \arcsin x dx = x \arcsin x + \sqrt {1 - x^2} +C$
$\int \arccos x dx = x \arccos x - \sqrt {1 - x^2} +C$
$\int \arctan x dx = x \arctan x - \ln \sqrt {(1 + x^2)} +C$
$\int \arccot x dx = x \arccot x + \ln \sqrt {(1 + x^2)} +C$
$\int \arcsec x dx = x \arcsec x - \ln (x - \sqrt{x^2 - 1}) +C$
$\int \arccsc x dx = x \arccsc x + \ln (x + \sqrt{x^2 - 1}) +C$

## 含有指数函数的积分

$\int e^x\mbox{d}x=e^x+C$
$\int\alpha^x\mbox{d}x=\frac{\alpha^x}{\ln\alpha}+C$
$\int xe^{ax}\mbox{d}x=\frac{1}{a^2}(ax-1)e^{ax}+C$
$\int x^ne^{ax}\mbox{d}x=\frac{1}{a}x^ne^{ax}-\frac{n}{a}\int x^{n-1}e^{ax}\mbox{d}x$
$\int e^{ax}\sin bx \mbox{d}x=\frac{e^{ax}}{a^2+b^2}(a\sin bx-b\cos bx)+C$
$\int e^{ax}\cos bx \mbox{d}x=\frac{e^{ax}}{a^2+b^2}(a\cos bx+b\sin bx)+C$

## 含有对数函数的积分

$\int\ln x\mbox{d}x = x\ln x - x + C$
$\int\log_\alpha x\mbox{d}x=\frac{1}{\ln\alpha}\left({x\ln x - x}\right)+C$
$\int x^n\ln x\mbox{d}x = \frac{x^{n+1}}{(n+1)^2}[(n+1)\ln x -1]+ C$
$\int\frac{1}{x\ln{x}}\mbox{d}x = \ln{(\ln{x})}+C$

## 含有双曲函数的积分

$\int \sinh x \mbox{d}x = \cosh x +C$
$\int \cosh x \mbox{d}x = \sinh x +C$
$\int \tanh x \mbox{d}x = \ln\cosh x +C$
$\int \coth x \mbox{d}x = \ln\left(\sinh x\right) +C$
$\int \mbox{sech}\ x \mbox{d}x = \arcsin (\tanh x) + C = \arctan (\sinh x) + C$
$\int \mbox{csch}\ x \mbox{d}x = \ln\left( \tanh {x \over2}\right) + C$

## 定积分

$\int^\infty_{-\infty}e^{-\alpha x^2}\mbox{d}x=\sqrt{\frac{\pi}{\alpha}}$
$\int_0^\frac{\pi}{2} \mbox{sin}^n x\mbox{d}x=\int_0^\frac{\pi}{2} \mbox{cos}^n x\mbox{d}x= \begin{cases} \frac{n-1}{n}\cdot\frac{n-3}{n-2}\cdot\cdots\cdot\frac{4}{5}\cdot\frac{2}{3}, & \mbox{if }n>1\mbox{ and }n\mbox{ is odd} \\ \frac{n-1}{n}\cdot\frac{n-3}{n-2}\cdot\cdots\cdot\frac{3}{4}\cdot\frac{1}{2}\cdot\frac{\pi}{2}, & \mbox{if }n>0\mbox{ and }n\mbox{ is even} \end{cases}$