# 积分列表

## 不定积分

### 多项式、指数及对数

• ${\displaystyle \int {c\;dx}=cx}$
• ${\displaystyle \int {x^{c}dx}={{x^{c+1}} \over {c+1}}}$  (若 c ≠- 1)
• ${\displaystyle \int {{1 \over x}dx}=\ln \left|x\right|}$

• ${\displaystyle \int {e^{x}dx}=e^{x}}$
• ${\displaystyle \int {c^{x}dx}={{c^{x}} \over {\ln c}}}$
• ${\displaystyle \int {\ln x\;dx}=x\left({\ln x-1}\right)}$
• ${\displaystyle \int {\log _{c}x\;dx}={{x\left({\ln x-1}\right)} \over {\ln c}}}$
• ${\displaystyle \int {x^{n}e^{-x}dx}=-[x^{n}+nx^{n-1}+n(n-1)x^{n-2}+n(n-1)(n-2)x^{n-3}+\cdots +n!x^{0}]e^{-x},n\in N}$

### 三角函数及反三角函数

• ${\displaystyle \int {\sin x\;dx}=-\cos x}$
• ${\displaystyle \int {\cos x\;dx}=\sin x}$
• ${\displaystyle \int {\tan x\;dx}=\ln \left|{\sec x}\right|}$
• ${\displaystyle \int {\cot x\;dx}=\ln \left|{\sin x}\right|}$
• ${\displaystyle \int {\sec x\;dx}=\ln \left|{\sec x+\tan x}\right|}$
• ${\displaystyle \int {\csc x\;dx}=\ln \left|{\tan {x \over 2}}\right|}$

• ${\displaystyle \int {\sin ^{2}x\;dx}={{2x-\sin 2x} \over 4}}$
• ${\displaystyle \int {\cos ^{2}x\;dx}={{2x+\sin 2x} \over 4}}$
• ${\displaystyle \int {\tan ^{2}x\;dx}=\tan x-x}$
• ${\displaystyle \int {\cot ^{2}x\;dx}=-\cot x-x}$
• ${\displaystyle \int {\sec ^{2}x\;dx}=\tan x}$
• ${\displaystyle \int {\csc ^{2}x\;dx}=-\cot x}$

• ${\displaystyle \int {\sin ^{3}x\;dx}={{\cos ^{3}x-3\cos x} \over 3}}$
• ${\displaystyle \int {\cos ^{3}x\;dx}={{3\sin x-\sin ^{3}x} \over 3}}$
• ${\displaystyle \int {\tan ^{3}x\;dx}={{\sec ^{2}x-\ln \sec ^{2}x} \over 2}}$
• ${\displaystyle \int {\cot ^{3}x\;dx}={{\ln \csc ^{2}x-\csc ^{2}x} \over 2}}$

• ${\displaystyle \int {\sin ^{-1}x\;dx}=x\sin ^{-1}x+{\sqrt {1-x^{2}}}}$
• ${\displaystyle \int {\cos ^{-1}x\;dx}=x\cos ^{-1}x-{\sqrt {1-x^{2}}}}$
• ${\displaystyle \int {\tan ^{-1}x\;dx}=x\tan ^{-1}x-\ln {\sqrt {1+x^{2}}}}$
• ${\displaystyle \int {\cot ^{-1}x\;dx}=x\cot ^{-1}x+\ln {\sqrt {1+x^{2}}}}$

### 双曲函数及反双曲函数

• ${\displaystyle \int {\sinh x\;dx}=\cosh x}$
• ${\displaystyle \int {\cosh x\;dx}=\sinh x}$
• ${\displaystyle \int {\tanh x\;dx}=\ln {\mathop {\rm {sech}} }x}$
• ${\displaystyle \int {\coth x\;dx}=\ln \left|{\sinh x}\right|}$
• ${\displaystyle \int {{\mathop {\rm {sech}} }x\;dx}=\sin ^{-1}\tanh x=2\tan ^{-1}e^{x}}$
• ${\displaystyle \int {{\mathop {\rm {csch}} }x\;dx}=\ln \left|{\tanh {x \over 2}}\right|=-2\coth ^{-1}e^{\left|x\right|}}$

• ${\displaystyle \int {\sinh ^{-1}x\;dx}=x\sinh ^{-1}x-{\sqrt {x^{2}+1}}}$
• ${\displaystyle \int {\cosh ^{-1}x}=x\cosh ^{-1}x\mp {\sqrt {x^{2}-1}}}$

### 其它形式

• ${\displaystyle \int {{dx} \over {n^{2}x^{2}+c^{2}}}={1 \over {nc}}{\tan ^{-1}{{nx} \over c}}}$
• ${\displaystyle \int {{dx} \over {n^{2}x^{2}-c^{2}}}=-{1 \over {nc}}{\tanh ^{-1}{{nx} \over c}}}$

• ${\displaystyle \int {{dx} \over {\sqrt {x^{2}+c^{2}}}}=\sinh ^{-1}{x \over c}}$
• ${\displaystyle \int {{dx} \over {\sqrt {x^{2}-c^{2}}}}=\ln \left({x+{\sqrt {x^{2}-c^{2}}}}\right)}$
• ${\displaystyle \int {{dx} \over {\sqrt {c^{2}-x^{2}}}}=\sin ^{-1}{x \over c}}$

• ${\displaystyle \int {{dx} \over {x{\sqrt {x^{2}+c^{2}}}}}=-{1 \over c}\ln \left({{c+{\sqrt {c^{2}+x^{2}}}} \over x}\right)}$
• ${\displaystyle \int {{dx} \over {x{\sqrt {x^{2}-c^{2}}}}}=-{1 \over c}\sec ^{-1}{x \over c}}$
• ${\displaystyle \int {{dx} \over {x{\sqrt {c^{2}-x^{2}}}}}=-{1 \over c}\ln \left({{c+{\sqrt {c^{2}-x^{2}}}} \over x}\right)}$

### 分部积分法

• ${\displaystyle \int {fg'\;dx}=fg-\int {f'g\;dx}}$

## 定积分

• ${\displaystyle \int _{0}^{+\infty }{\frac {\sin x}{x}}dx={\frac {\pi }{2}}}$
• ${\displaystyle \int _{0}^{+\infty }{\frac {\sin ^{2}x}{x^{2}}}dx={\frac {\pi }{2}}}$
• ${\displaystyle \int _{0}^{+\infty }x^{z}e^{-\lambda x}dx=\lambda ^{-(z+1)}\Gamma (z+1)}$  (其中${\displaystyle \lambda >0}$ )
• ${\displaystyle \int _{-1}^{1}e^{izx}dx={\frac {2\sin z}{z}}}$
• ${\displaystyle \int _{0}^{\pi /2}\cos ^{a}x\ \sin ^{b}x\ dx={\frac {1}{2}}B({\frac {a+1}{2}},{\frac {b+1}{2}})}$  （其中${\displaystyle a,b>-1}$ , ${\displaystyle B(x,y)={\frac {\Gamma (x)\Gamma (y)}{\Gamma (x+y)}}}$