積分表

由於列表比較長，積分表被分為以下幾個部分：

- 有理函數積分表
- 無理函數積分表
- 指數函數積分表
- 對數函數積分表
- 高斯函數積分表
- 三角函數積分表
- 反三角函數積分表
- 雙曲函數積分表
- 反雙曲函數積分表

## 含有$ax+b$的積分

$$\int\ (ax+b)^n\mbox{d}x=\frac{(ax+b)^{n+1}}{a(n+1)}+C,n\neq -1$$

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

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

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

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

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

## 含有$\sqrt{a+bx}$的積分

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

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

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

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

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

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

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

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

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

## 含有$x^2\pm\alpha^2$的積分

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

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

## 含有 ${ax^2+b}$的積分

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

## 含有 $ax^2+bx+c$的積分

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

## 含有 $\sqrt{a^2+x^2}\qquad(a>0)$的積分

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

## 含有$R=\sqrt{|a|x^2+bx+c}\qquad(a\ne0)$的積分

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

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

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

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

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

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

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

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

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

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

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

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

## 含有三角函數的積分

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

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

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

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

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

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

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

$$\int\cot x\mbox{d}x=\ln{\left|\sin x\right|}+C$$

$$\int\sec x\mbox{d}x=\ln{\left|\sec x+\tan x\right|}+C$$

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

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

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

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

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

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

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

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

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

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

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

## 含有反三角函數的積分

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

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

$$\int \arctan{x} \, dx = x \arctan{x} - \frac{1}{2} \ln { \vert 1 + x^2 \vert } + C$$

$$\int \arccot{x} \, dx = x \arccot{x} + \frac{1}{2} \ln { \vert 1 + x^2 \vert } + C$$

$$\int \arcsec x \mbox{d}x = x \arcsec x - \sgn(x)\ln \left|x + \sqrt{x^2 - 1}\right| +C <pre> = x \arcsec x + \sgn(x)\ln \left|x - \sqrt{x^2 - 1}\right| +C$$

$$\int \arccsc x \mbox{d}x = x \arccsc x + \sgn(x)\ln \left|x + \sqrt{x^2 - 1}\right| +C <pre> = x \arccsc x - \sgn(x)\ln \left|x - \sqrt{x^2 - 1}\right| +C$$

## 含有指數函數的積分

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

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

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

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

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

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

## 含有對數函數的積分

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

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

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

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

## 含有雙曲函數的積分

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

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

$$\int \tanh x \mbox{d}x = \ln\left(\cosh x\right) +C$$

$$\int \coth x \mbox{d}x = \ln\left|\sinh x\right| +C$$

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

$$\int \mbox{csch}\ x \mbox{d}x = \ln\left|\tanh {x \over2}\right| + C$$

## 定積分

高斯積分

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

沃利斯乘積

$\int_0^\frac{\pi}{2} \mbox{sin}^n x\mbox{d}x=\int_0^\frac{\pi}{2} \mbox{cos}^n x\mbox{d}x= \begin{cases} \frac{n-1}{n}\cdot\frac{n-3}{n-2}\cdot\ldots\cdot\frac{4}{5}\cdot\frac{2}{3}, & \mbox{if }n>1\mbox{ 且 }n\mbox{為 奇 數 } \\ \frac{n-1}{n}\cdot\frac{n-3}{n-2}\cdot\ldots\cdot\frac{3}{4}\cdot\frac{1}{2}\cdot\frac{\pi}{2}, & \mbox{if }n>0\mbox{ 且 }n\mbox{為 偶 數 } \end{cases}$