# 定积分公式

## 定积分公式

WARNING

### 基本积分表

1. $\int\frac{1}{x} dx=\ln|x|+C$

2. $\int\frac{1}{1+x^2} dx=\arctan{x}+C$

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

4. $\int\cos{x} dx=\sin{x}+C$

5. $\int\sin{x} dx=-\cos{x}+C$

6. $\int\frac{1}{\cos{x^2}} dx=\int\sec^2x dx=\tan x +C$

7. $\int\frac{1}{sin^2x} dx=\int\csc^2x dx=-\cot x+C$

8. $\int\sec x \tan x dx=\sec x+C$

9. $\int\csc x \cot x dx=-\csc x+C$

10. $\int e^x dx=e ^x+C$

11. $\int a^x dx=\frac{a^x}{\ln a}+C$

12. $\int \frac {1}{\sqrt{x}} dx=2\sqrt{x}+C$

### 第一类换元法

1. $\int f(ax+b) dx=\frac 1 a \int f(ax+b)d(ax+b)$

2. $\int f(x^n)x^{n-1} dx=\frac 1 n \int f(x^n)d(x^n)$

3. $\int f(x^n)\frac 1 x dx=\frac 1 n \int f(x^n)\frac 1 {x^n}d(x^n)$

4. $\int f(\sin x)\cos x dx=\int f(\sin x)d\sin x$

5. $\int f(\cos x)\sin x dx=-\int f(\cos x)d \cos x$

6. $\int f(\tan x) \sec^2 x dx=\int f(\tan x) d\tan x$

7. $\int f(e^x)e^x dx=\int f(e^x) de^x$

8. $\int f(\ln x)\frac 1 x dx=\int f(\ln x) d\ln x$

9. $\int f(\arctan x) \frac 1 {1+x^2} dx=\int f(\arctan x) d\arctan x$

10. $\int f(\arcsin x) \frac 1 {\sqrt {1-x^2}} dx=\int f(\arcsin x) d\arcsin x$

1. $\int \tan x dx=-\ln |\cos x|+C$

2. $\int \cot x dx=\ln |\sin x|+C$

3. $\int \sec x dx=\ln |\sec x +\tan x|+C$

4. $\int \csc x dx=\ln |\csc x -\cot x|+C$

### 第二类换元法

Q:$\int f(x)=\sqrt[n]{ax+b} dx$

A:令$t=\sqrt[n]{ax+b}$

Q:$\int f(x)=\sqrt[n]\frac {ax+b}{cx+d} dx$

A:令$t=\sqrt[n]\frac {ax+b}{cx+d}$

Q:$\int f(x)=\sqrt{a^2-x^2} dx$

A:令$x=a\sin{t}$ $t\in(-\frac{\pi}{2},\frac{\pi}{2})$

Q:$\int f(x)=\sqrt{a^2+x^2} dx$

A:令$x=a\tan t$ $t\in(-\frac{\pi}{2},\frac{\pi}{2})$

Q:$\int f(x)=\sqrt{x^2-a^2} dx$

A:令$x=a\sec t$ $t\in(0,\frac{\pi}{2})$

Q:$\int f(x)=a^x dx$

A:令$x=a^x$

1. $\int\frac{1}{x^2+a^2} dx=\frac{1}{a}\arctan \frac{x}{a}+C$

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

3. $\int\frac{1}{\sqrt{x^2 \pm a^2}} dx=\ln{|x+\sqrt{x^2 \pm a^2}|}+C$

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

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

6. $\int\sqrt{a^2+x^2} dx=\frac{a^2}{2}\ln(x+\sqrt{a^2+x^2})+\frac{x}{2}\sqrt{a^2+x^2}+C$

### 参考资料

1. 同济大学数学系.高等数学上册[M].第7版.北京:高等教育出版社