积分表

Template:微积分学 Template:TOCright 由于列表比较长，积分表被分为以下几个部分：

• 有理函数积分表
• 无理函数积分表
• 指数函数积分表
• 对数函数积分表
• 高斯函数积分表
• 三角函数积分表
• 反三角函数积分表
• 双曲函数积分表
• 反双曲函数积分表

含有$ax+b$的积分

$\int (ax+b{)}^n\text{d}x=\frac{(ax+b{)}^{n+1}}{a(n+1)}+C$

$\int \frac{1}{ax+b}\text{d}x=\frac{1}{a}\ln |ax+b|+C$

$\int \frac{x}{ax+b}\text{d}x=\frac{1}{a^2}(ax+b-b\ln |ax+b|)+C$

$\int \frac{x^2}{ax+b}\text{d}x=\frac{1}{2a^3}[(ax+b{)}^2-4b(ax+b)+2b^2\ln |ax+b|]+C$

$\int \frac{1}{x(ax+b)}\text{d}x=-\frac{1}{b}\ln \left|\frac{ax+b}{x}\right|+C$

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

含有$\sqrt{a+bx}$的积分

$\int x\sqrt{a+bx}\text{d}x=\frac{2}{15b^2}(3bx-2a)(a+bx{)}^{\frac{3}{2}}+C$

$\int x^2\sqrt{a+bx}\text{d}x=\frac{2}{105b^3}(15b^2{x}^2-12abx+8a^2)(a+bx{)}^{\frac{3}{2}}+C$

$\int x^n\sqrt{a+bx}\text{d}x=\frac{2}{b(2n+3)}{x}^n(a+bx{)}^{\frac{3}{2}}-\frac{2na}{b(2n+3)}\int x^{n-1}\sqrt{a+bx}\text{d}x$

$\int \frac{\sqrt{a+bx}}{x}\text{d}x=2\sqrt{a+bx}+a\int \frac{1}{x\sqrt{a+bx}}\text{d}x$

$\int \frac{\sqrt{a+bx}}{x^n}\text{d}x=\frac{-1}{a(n-1)}\frac{(a+bx{)}^{\frac{3}{2}}}{x^{n-1}}-\frac{(2n-5)b}{2a(n-1)}\int \frac{\sqrt{a+bx}}{x^{n-1}}\text{d}x,n\ne 1$

$\int \frac{1}{x\sqrt{a+bx}}\text{d}x=\frac{1}{\sqrt{a}}\ln \left(\frac{\sqrt{a+bx}-\sqrt{a}}{\sqrt{a+bx}+\sqrt{a}}\right)+C,a>0$

$=\frac{2}{\sqrt{-a}}\arctan \sqrt{\frac{a+bx}{-a}}+C,a<0$

$\int \frac{x}{\sqrt{a+bx}}\text{d}x=\frac{2(a+bx{)}^{\frac{3}{2}}}{3b^2}-\frac{(2a)\sqrt{a+bx}}{b^2}$

$\int \frac{1}{x^n\sqrt{a+bx}}\text{d}x=\frac{-1}{a(n-1)}\frac{\sqrt{a+bx}}{x^{n-1}}-\frac{(2n-3)b}{2a(n-1)}\int \frac{1}{x^{n-1}}\sqrt{a+bx}\text{d}x,n\ne 1$

含有$x^2\pm {\alpha }^2$的积分

$\int \frac{1}{x^2+{\alpha }^2}\text{d}x=\frac{\arctan \frac{x}{\alpha }}{\alpha }+C$

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

含有 $a{x}^2+b$的积分

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

含有 $a{x}^2+bx+c$的积分

$\int (a{x}^2+bx+c)\text{d}x=\frac{a{x}^3}{3}+\frac{b{x}^2}{2}+cx+C$

含有 $\sqrt{a^2+x^2}(a>0)$的积分

$\int \sqrt{a^2+x^2}\text{d}x=\frac{1}{2}x\sqrt{a^2+x^2}+\frac{1}{2}{a}^2\ln (x+\sqrt{a^2+x^2})+C$

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

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

$\int \frac{\sqrt{a^2+x^2}}{x^2}\text{d}x=\ln (x+\sqrt{a^2+x^2})-\frac{\sqrt{a^2+x^2}}{x}+C$

$\int \frac{1}{\sqrt{a^2+x^2}}\text{d}x=\ln (x+\sqrt{a^2+x^2})+C$

$\int \frac{x^2}{\sqrt{a^2+x^2}}\text{d}x=\frac{1}{2}x\sqrt{a^2+x^2}-\frac{1}{2}{a}^2\ln (\sqrt{a^2+x^2}+x)+C$

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

$\int \frac{1}{x^2\sqrt{a^2+x^2}}\text{d}x=-\frac{\sqrt{a^2+x^2}}{a^{2}x}+C$

含有$\sqrt{x^2-a^2}(x^2>a^2)$的积分

$\int \frac{1}{\sqrt{x^2-a^2}}\text{d}x=ln|x+\sqrt{x^2-a^2}|+C$

含有$\sqrt{a^2-x^2}(a^2>x^2)$的积分

$\int \sqrt{a^2-x^2}\text{d}x=\frac{1}{2}x\sqrt{a^2-x^2}+\frac{a^2}{2}\arcsin \frac{x}{a}+C$

$\int \frac{1}{\sqrt{a^2-x^2}}\text{d}x=\arcsin \frac{x}{a}+C=-\arccos \frac{x}{a}+C$

$\int x^2\sqrt{a^2-x^2}\text{d}x=\frac{1}{8}x(2x^2-a^2)\sqrt{a^2-x^2}+\frac{1}{8}{a}^4\arcsin \frac{x}{a}+C$

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

$\int \frac{\sqrt{a^2-x^2}}{x^2}\text{d}x=-\frac{\sqrt{a^2-x^2}}{x}-\arcsin \frac{x}{a}+C$

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

$\int \frac{x^2}{\sqrt{a^2-x^2}}\text{d}x=-\frac{1}{2}x\sqrt{a^2-x^2}+\frac{1}{2}{a}^2\arcsin \frac{x}{a}+C$

$\int \frac{1}{x^2\sqrt{a^2-x^2}}\text{d}x=-\frac{\sqrt{a^2-x^2}}{a^{2}x}+C$

含有$R=\sqrt{|a|x^2+bx+c}(a\ne 0)$的积分

$\int \frac{\text{d}x}{R}=\frac{1}{\sqrt{a}}\ln (2\sqrt{a}R+2ax+b)(\text{for }a>0)$

$\int \frac{\text{d}x}{R}=\frac{1}{\sqrt{a}}\mathrm{arsinh}\frac{2ax+b}{\sqrt{4ac-b^2}}\text{(for }a>0\text{, }4ac-b^2>0\text{)}$

$\int \frac{\text{d}x}{R}=\frac{1}{\sqrt{a}}\ln |2ax+b|\text{(for }a>0\text{, }4ac-b^2=0\text{)}$

$\int \frac{\text{d}x}{R}=-\frac{1}{\sqrt{-a}}\arcsin \frac{2ax+b}{\sqrt{b^2-4ac}}\text{(for }a<0\text{, }4ac-b^2<0\text{, }(2ax+b)<\sqrt{b^2-4ac}\text{)}$

$\int \frac{\text{d}x}{R^3}=\frac{4ax+2b}{(4ac-b^2)R}$

$\int \frac{\text{d}x}{R^5}=\frac{4ax+2b}{3(4ac-b^2)R}(\frac{1}{R^2}+\frac{8a}{4ac-b^2})$

$\int \frac{\text{d}x}{R^{2n+1}}=\frac{2}{(2n-1)(4ac-b^2)}[\frac{2ax+b}{R^{2n-1}}+4a(n-1)\int \frac{\text{d}x}{R^{2n-1}}]$

$\int \frac{x}{R}\text{d}x=\frac{R}{a}-\frac{b}{2a}\int \frac{\text{d}x}{R}$

$\int \frac{x}{R^3}\text{d}x=-\frac{2bx+4c}{(4ac-b^2)R}$

$\int \frac{x}{R^{2n+1}}\text{d}x=-\frac{1}{(2n-1)a{R}^{2n-1}}-\frac{b}{2a}\int \frac{\text{d}x}{R^{2n+1}}$

$\int \frac{\text{d}x}{xR}=-\frac{1}{\sqrt{c}}\ln \left(\frac{2\sqrt{c}R+bx+2c}{x}\right)$

$\int \frac{\text{d}x}{xR}=-\frac{1}{\sqrt{c}}\mathrm{arsinh}\left(\frac{bx+2c}{|x|\sqrt{4ac-b^2}}\right)$

含有三角函数的积分

$\int \cos x\text{d}x=\sin x+C$

$\int \sin x\text{d}x=-\cos x+C$

$\int {\sec }^{2}x\text{d}x=\tan x+C$

$\int {\csc }^{2}x\text{d}x=-\cot x+C$

$\int \sec x\tan x\text{d}x=\sec x+C$

$\int \csc x\cot x\text{d}x=-\csc x+C$

$\int \tan x\text{d}x=-\ln |\cos x|+C=\ln |\sec x|+C$

$\int \cot x\text{d}x=\ln |\sin x|+C$

$\int \sec x\text{d}x=\ln |\sec x+\tan x|+C$

$\int \csc x\text{d}x=\ln |\csc x-\cot x|+C=\ln \left|\frac{\tan x-\sin x}{\sin x\tan x}\right|+C$

$\int {\sin }^{n}x\text{d}x=-\frac{1}{n}{\sin }^{n-1}x\cos x+\frac{n-1}{n}\int {\sin }^{n-2}x\text{d}x+C\mathrm{\forall }n\ge 2$

$\int {\sin }^{2}x\text{d}x=\frac{x}{2}-\frac{\sin 2x}{4}+C$

$\int {\cos }^{n}x\text{d}x=\frac{1}{n}{\cos }^{n-1}x\sin x+\frac{n-1}{n}\int {\cos }^{n-2}x\text{d}x+C\mathrm{\forall }n\ge 2$

$\int {\cos }^{2}x\text{d}x=\frac{x}{2}+\frac{\sin 2x}{4}+C$

$\int {\tan }^{n}x\text{d}x=\frac{1}{n-1}{\tan }^{n-1}x-\int {\tan }^{n-2}x\text{d}x+C\mathrm{\forall }n\ge 2$

$\int {\tan }^{2}x\text{d}x=\tan x-x+C$

$\int {\cot }^{n}x\text{d}x=-\frac{1}{n-1}{\cot }^{n-1}x-\int {\cot }^{n-2}x\text{d}x+C\mathrm{\forall }n\ge 2$

$\int {\cot }^{2}x\text{d}x=-\cot x-x+C$

$\int {\sec }^{n}x\text{d}x=\frac{1}{n-1}{\sec }^{n-2}x\tan x+\frac{n-2}{n-1}\int {\sec }^{n-2}x\text{d}x+C\mathrm{\forall }n\ge 2$

$\int {\csc }^{n}x\text{d}x=-\frac{1}{n-1}{\csc }^{n-2}x\cot x+\frac{n-2}{n-1}\int {\csc }^{n-2}x\text{d}x+C\mathrm{\forall }n\ge 2$

含有反三角函数的积分

$\int \arcsin x\text{d}x=x\arcsin x+\sqrt{1-x^2}+C$

$\int \arccos x\text{d}x=x\arccos x-\sqrt{1-x^2}+C$

$\int \arctan xdx=x\arctan x-\frac{1}{2}\ln |1+x^2|+C$

$\int \mathrm{arccot}xdx=x\mathrm{arccot}x+\frac{1}{2}\ln |1+x^2|+C$

$\int \mathrm{arcsec}x\text{d}x=x\mathrm{arcsec}x-\mathrm{sgn}(x)\ln |x+\sqrt{x^2-1}|+C=x\mathrm{arcsec}x+\mathrm{sgn}(x)\ln |x-\sqrt{x^2-1}|+C$

$\int \mathrm{arccsc}x\text{d}x=x\mathrm{arccsc}x+\mathrm{sgn}(x)\ln |x+\sqrt{x^2-1}|+C=x\mathrm{arccsc}x-\mathrm{sgn}(x)\ln |x-\sqrt{x^2-1}|+C$

含有指数函数的积分

$\int e^x\text{d}x=e^x+C$

$\int {\alpha }^x\text{d}x=\frac{{\alpha }^x}{\ln \alpha }+C$

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

$\int x^n{e}^{ax}\text{d}x=\frac{1}{a}{x}^n{e}^{ax}-\frac{n}{a}\int x^{n-1}{e}^{ax}\text{d}x$

$\int e^{ax}\sin bx\text{d}x=\frac{e^{ax}}{a^2+b^2}(a\sin bx-b\cos bx)+C$

$\int e^{ax}\cos bx\text{d}x=\frac{e^{ax}}{a^2+b^2}(a\cos bx+b\sin bx)+C$

含有对数函数的积分

$\int \ln x\text{d}x=x\ln x-x+C$

$\int {\log }_{\alpha }x\text{d}x=\frac{1}{\ln \alpha }\left(x\ln x-x\right)+C$

$\int x^n\ln x\text{d}x=\frac{x^{n+1}}{(n+1{)}^2}[(n+1)\ln x-1]+C$

$\int \frac{1}{x\ln x}\text{d}x=\ln (\ln x)+C$

含有双曲函数的积分

$\int \sinh x\text{d}x=\cosh x+C$

$\int \cosh x\text{d}x=\sinh x+C$

$\int \tanh x\text{d}x=\ln (\cosh x)+C$

$\int \coth x\text{d}x=\ln |\sinh x|+C$

$\int \text{sech}x\text{d}x=\arcsin (\tanh x)+C=\arctan (\sinh x)+C$

$\int \text{csch}x\text{d}x=\ln |\tanh \frac{x}{2}|+C$

定積分

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{-\alpha x^2}\text{d}x=\sqrt{\frac{\pi }{\alpha }}$

$\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\text{sin}}^{n}x\text{d}x=\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\text{cos}}^{n}x\text{d}x=\{\begin{array}{cc}\frac{n-1}{n}\cdot \frac{n-3}{n-2}\cdot \dots \cdot \frac{4}{5}\cdot \frac{2}{3},& \text{if }n>1\text{ 且 }n\text{為 奇 數 }\\ \frac{n-1}{n}\cdot \frac{n-3}{n-2}\cdot \dots \cdot \frac{3}{4}\cdot \frac{1}{2}\cdot \frac{\pi }{2},& \text{if }n>0\text{ 且 }n\text{為 偶 數 }\end{array}$