积分表

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
• 三角函数积分表
• 指数函数积分表
• 对数函数积分表
• 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 a + bx 的积分

$\int\frac{1}{a+bx}dx=\frac{1}{b}\ln \left | a+bx\right|+C$

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

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

$\int\frac{1}{x(a+bx)}dx = \frac{-1}{a}\ln\left | \frac{a+bx}{x}\right |+C$

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

含有√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 ± α^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 ax² + b （a>0）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 √a² + x² （a>0）的积分

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

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

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

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

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

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

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

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

含有 √x² − a² 的积分

对于 $x^2>a^2$

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

含有 √a² − x² (a>0) 的积分

对于 $a^2>x^2$

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

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

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

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

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

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

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

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

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

含有$\sqrt{|a|x^2+bx+c}\qquad(\mbox{for }a\ne0)$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

$\int^\infty_{-\infty}e^{-\alpha x^2}\mbox{d}x=\sqrt{\frac{\pi}{\alpha}}$