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
• 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\mbox{d}x=\frac{(ax+b)^{n+1}}{a(n+1)}+C,n\neq -1
:\int\frac{1}{ax+b}\mbox{d}x=\frac{1}{a}\ln \left | ax+b\right|+C
:\int\frac{x}{ax+b}\mbox{d}x=\frac{1}{a^2}(ax+b-b\ln\left |ax+b \right|) +C
:\int\frac{x^2}{ax+b}\mbox{d}x=\frac{1}{2a^3} \left[(ax+b)^2-4b(ax+b)+2b^2 \ln\left |ax+b\right| \right]+C
:\int\frac{1}{x(ax+b)}\mbox{d}x = -\frac{1}{b}\ln\left | \frac{ax+b}{x}\right |+C
:\int\frac{1}{x^2(ax+b)}\mbox{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}\mbox{d}x=\frac{2}{15b^2}(3bx-2a)(a+bx)^{\frac{3}{2}}+C
:\int x^2\sqrt{a+bx}\mbox{d}x=\frac{2}{105b^3}(15b^2x^2-12abx+8a^2)(a+bx)^{\frac{3}{2}}+C
:\int x^n\sqrt{a+bx}\mbox{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}\mbox{d}x
:\int\frac{\sqrt{a+bx}}{x}\mbox{d}x=2\sqrt{a+bx}+a\int\frac{1}{x\sqrt{a+bx}}\mbox{d}x
:\int\frac{\sqrt{a+bx}}{x^n}\mbox{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}}\mbox{d}x,n\neq 1
:\int\frac{1}{x\sqrt{a+bx}}\mbox{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
:\int\frac{x}{\sqrt{a+bx}}\mbox{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}}\mbox{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}\mbox{d}x,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}\mbox{d}x=\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}\mbox{d}x=\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}\mbox{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}\mbox{d}x = \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}}\mbox{d}x=\ln \left ( x+\sqrt{a^2+x^2} \right ) +C
:\int\frac{x^2}{\sqrt{a^2+x^2}}\mbox{d}x=\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}}\mbox{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}}\mbox{d}x=-\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}}\mbox{d}x=\ln| x+\sqrt{x^2-a^2} | +C

含有\sqrt{a^2-x^2} \qquad(a^2>x^2)的积分
:\int\sqrt{a^2-x^2}\mbox{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}}\mbox{d}x= \arcsin \frac{x}{a} +C = - \arccos \frac{x}{a} +C
:\int x^2\sqrt{a^2-x^2}\mbox{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}\mbox{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}\mbox{d}x=-\frac{\sqrt{a^2-x^2}}{x} -\arcsin\frac{x}{a}+C
:\int\frac{1}{x\sqrt{a^2-x^2}}\mbox{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}}\mbox{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}}\mbox{d}x=-\frac{\sqrt{a^2-x^2}}{a^2x}+C

==含有R=\sqrt+C=\ln{\left|\sec x\right|}+C
:\int\cot x\mbox{d}x=\ln{\left|\sin x\right|}+C
:\int\sec x\mbox{d}x=\ln{\left|\sec x+\tan x\right|}+C
:\int\csc x\mbox{d}x=\ln{\left|\csc x-\cot x\right|}+C=\ln{\left|{\tan x-\sin x\over\sin x\tan x}\right|}+C

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

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

:\int \tan ^n x \mbox{d}x = \frac{1}{n-1} \tan ^{n-1} x - \int \tan ^{n-2} x \mbox{d}x +C \quad \forall n \ge 2
:\int \tan ^2 x \mbox{d}x = \tan x - x +C

:\int \cot ^n x \mbox{d}x = - \frac{1}{n-1} \cot ^{n-1} x - \int \cot ^{n-2} x \mbox{d}x +C \quad \forall n \ge 2
:\int \cot ^2 x \mbox{d}x = - \cot x - x +C

:\int \sec ^n x \mbox{d}x = \frac{1}{n-1} \sec ^{n-2} x \tan x + \frac{n-2}{n-1} \int \sec ^{n-2} x \mbox{d}x +C \quad \forall n \ge 2

:\int \csc ^n x \mbox{d}x = - \frac{1}{n-1} \csc ^{n-2} x \cot x + \frac{n-2}{n-1} \int \csc ^{n-2} x \mbox{d}x +C \quad \forall n \ge 2

含有反三角函数的积分
:\int \arcsin x \mbox{d}x = x \arcsin x + \sqrt {1 - x^2} +C
:\int \arccos x \mbox{d}x = x \arccos x - \sqrt {1 - x^2} +C
:\int \arctan{x} \, dx = x \arctan{x} - \frac{1}{2} \ln { \vert 1 + x^2 \vert } + C
:\int \arccot{x} \, dx = x \arccot{x} + \frac{1}{2} \ln { \vert 1 + x^2 \vert } + C
:\int \arcsec x \mbox{d}x = x \arcsec x - \sgn(x)\ln \left|x + \sqrt{x^2 - 1}\right| +C
= x \arcsec x + \sgn(x)\ln \left|x - \sqrt{x^2 - 1}\right| +C
:\int \arccsc x \mbox{d}x = x \arccsc x + \sgn(x)\ln \left|x + \sqrt{x^2 - 1}\right| +C
= x \arccsc x - \sgn(x)\ln \left|x - \sqrt{x^2 - 1}\right| +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\left(\cosh x\right) +C
:\int \coth x \mbox{d}x = \ln\left|\sinh x\right| +C
:\int \mbox{sech}\ x \mbox{d}x = \arcsin\left(\tanh x\right) + C = \arctan\left(\sinh x\right) + C
:\int \mbox{csch}\ x \mbox{d}x = \ln\left|\tanh {x \over2}\right| + C

定積分
高斯積分
:\int^\infty_{-\infty}e^{-\alpha x^2}\mbox{d}x=\sqrt{\frac{\pi}{\alpha}}

沃利斯乘积
:\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\ldots\cdot\frac{4}{5}\cdot\frac{2}{3}, & \mbox{if }n>1\mbox{ 且 }n\mbox{為 奇 數 } \\
\frac{n-1}{n}\cdot\frac{n-3}{n-2}\cdot\ldots\cdot\frac{3}{4}\cdot\frac{1}{2}\cdot\frac{\pi}{2}, & \mbox{if }n>0\mbox{ 且 }n\mbox{為 偶 數 }
\end{cases}