Advanced Mathematics - Commonly Used Indefinite Integral Formulas

一、基本积分表

1. $\int kdx=kx+C$
2. $\int x^{u}dx=\text{u+1xu+1}+C$
3. $\int \frac{dx}{x}=ln|x|+C$
4. $\int \frac{dx}{1+x^2}=arctanx+C$
5. $\int \frac{dx}{\sqrt{1-x^2}}=arcsinx+C$
6. $\int cosxdx=sinx+C$
7. $\int sinxdx=-cosx+C$
8. $\int \frac{dx}{co{s}^{2}x}=\int se{c}^{2}xdx=tanx+C$
9. $\int \frac{dx}{si{n}^{2}x}=\int cs{c}^{2}xdx=cotx+C$
10. $\int secxtanxdx=secx+C$
11. $\int cscxcotxdx=cscx+C$
12. $\int e^{x}dx=e^x+C$
13. $\int a^{x}dx=\frac{a^x}{lna}+C$

二、special integral formula

1. $\int \text{sh}x\text{d}x=\text{ch}x+C$(Hyperbolic integral formula${\text{ch}}^{2}t-{\text{sh}}^{2}t=1$)
2. $\int \text{ch}x\text{d}x=\text{sh}x+C$(Hyperbolic integral formula${\text{ch}}^{2}t-{\text{sh}}^{2}t=1$)
3. $\int \text{tan}x\text{d}x=\text{–}\text{ln}|\text{cos}x|+C$
4. $\int \text{cot}x\text{d}x=\text{ln}|\text{sin}x|+C$
5. $\int \text{sec}x\text{d}x=\text{ln}|\text{sec}x+\text{tan}x|+C$
6. $\int \text{csc}x\text{d}x=\text{ln}|\text{csc}x-\text{cot}x|+C$
7. $\int \frac{dx}{a^2+x^2}=\frac{1}{a}\text{arctan}\frac{x}{a}+C$
8. $\int \frac{dx}{a^2-x^2}=\frac{1}{2a}\text{ln}|\frac{x-a}{x+a}|+C$
9. $\int \frac{dx}{\sqrt{a^2-x^2}}=\text{arcsin}\frac{x}{a}+C$
10. $\int \frac{dx}{\sqrt{x^2+a^2}}=\text{ln}|x+\sqrt{x^2+a^2}|+C$
11. $\int \frac{dx}{\sqrt{x^2-a^2}}=\text{ln}|x+\sqrt{x^2-a^2}|+C$

三、分部积分

$\int u{v}^{\prime }\text{d}x=uv-\int u^{\prime }v\text{d}x$
或者
$\int u\text{d}v=uv-\int v\text{d}u$