不定积分公式表

设 $k, \mu, a(a>0, a\not ={1})$ 是常数.

1. $\int k \ \mathrm{d}x=kx+C$
2. $\int\dfrac{1}{x}\mathrm{d}x=\ln{|x|}+C$
3. $\int x^{\mu}\ \mathrm{d}x=\dfrac{x^{\mu+1}}{\mu+1}+C$, 特例 $\int\dfrac{1}{x^2}\mathrm{d}x=-\dfrac{1}{x}+C, \int\dfrac{1}{\sqrt{x}}\mathrm{d}x=2\sqrt{x}+C$
4. $\int a^x\ \mathrm{d}x=\dfrac{a^x}{\ln{a}}+C$, 特例 $\int e^x\ \mathrm{d}x=e^x+C$
5. $\int\cos{x} \ \mathrm{d}x=\sin{x}+C$
6. $\int\sin{x} \ \mathrm{d}x=-\cos{x}+C$
7. $\int\dfrac{1}{\cos^2{x}}\mathrm{d}x=\int\sec^2{x}\ \mathrm{d}x=\tan{x}+C$
8. $\int\dfrac{1}{\sin^2{x}}\mathrm{d}x=\int\csc^2{x}\ \mathrm{d}x=-\cot{x}+C$
9. $\int\dfrac{1}{a^2+x^2}\mathrm{d}x=\dfrac{1}{a}\arctan{\dfrac{x}{a}}+C$, 特例 $\int\dfrac{1}{1+x^2}\mathrm{d}x=\arctan{x}+C$
10. $\int\dfrac{1}{a^2-x^2}\mathrm{d}x=\dfrac{1}{2a}\ln{\left|\dfrac{a+x}{a-x}\right|}+C$, 特例 $\int\dfrac{1}{1-x^2}\mathrm{d}x=\mathrm{arth}\ x+C$
11. $\int\dfrac{1}{\sqrt{a^2-x^2}}\mathrm{d}x=\arcsin{\dfrac{x}{a}}+C$, 特例 $\int\dfrac{1}{\sqrt{1-x^2}}\mathrm{d}x=\arcsin{x}+C$
12. $\int\dfrac{1}{\sqrt{x^2\pm a^2}}\mathrm{d}x=\ln{\left|x+\sqrt{x^2\pm a^2}\right|+C}$, 特例 $\int\dfrac{1}{\sqrt{x^2+1}}\mathrm{d}x=\mathrm{arsh}\ x+C$
13. $\int\ch{x}\ \mathrm{d}x=\sh{x}+C$
14. $\int\sh{x}\ \mathrm{d}x=\ch{x}+C$
15. $\int\sec{x}\ \mathrm{d}x=\ln{\left|\sec{x}+\tan{x}\right|}+C$
16. $\int\csc{x}\ \mathrm{d}x=\ln{\left|\csc{x}-\cot{x}\right|}+C$

其中,

$\sh{x}(\sinh{x})=\dfrac{e^x-e^{-x}}{2}$ 为双曲正弦函数

$\ch{x}(\cosh{x})=\dfrac{e^x+e^{-x}}{2}$ 为双曲余弦函数

$\th{x}(\tanh{x})=\dfrac{\sh{x}}{\ch{x}}=\dfrac{e^x-e^{-x}}{e^x+e^{-x}}$ 为双曲正切函数