基本积分表

24个基本积分：

①$\int kdx=kx+C$

②$\int x^{u}dx=\frac{x^{u+1}}{u+1}+C$

③$\int \frac{1}{x}dx=\ln |x|+C$

④$\int \frac{1}{1+x^2}dx=\arctan x+C=-arccotx+C$
⑤$\int \frac{1}{\sqrt{1-x^2}}=\arcsin x+C=-\arccos x+C$

⑥$\int \cos xdx=\sin x+C$
⑦$\int \sin xdx=-\cos x+C$

⑧$\int \frac{1}{co{s}^{2}x}dx=\int {\sec }^{2}xdx=\tan x+C$
⑨$\int \frac{1}{si{n}^{2}x}dx=\int {\csc }^{2}xdx=-\cot x+C$

⑩$\int \sec x\tan xdx=\sec x+C$
⑪$\int \csc x\cot xdx=-\csc x+C$

⑫$\int e^{x}dx=e^x+C$
⑬$\int a^{x}dx=\frac{a^x}{\ln a}+C$

⑭$\int shxdx=chx+C$
⑮$\int chxdx=shx+C$

⑯$\int \tan xdx=-\ln |\cos x|+C$
⑰$\int \cot xdx=\ln |\sin x|+C$
⑱$\int \sec xdx=\ln |\sec x+\tan x|+C$
⑲$\int \csc xdx=\ln |cscx-\cot x|+C$

⑳$\int \frac{1}{x^2+a^2}dx=\frac{1}{a}\arctan \frac{x}{a}+C$
㉑$\int \frac{1}{x^2-a^2}dx=\frac{1}{2a}\ln |\frac{x-a}{x+a}|+C$

㉒$\int \frac{1}{\sqrt{a^2-x^2}}dx=\arcsin \frac{x}{a}+C$
㉓$\int \frac{1}{\sqrt{x^2+a^2}}dx=\ln (x+\sqrt{x^2+a^2})+C$
㉔$\int \frac{1}{\sqrt{x^2-a^2}}dx=\ln |x+\sqrt{x^2-a^2}|+C$

两个由基本积分②推导的常用积分
①$\int \frac{1}{\sqrt{x}}dx=2\sqrt{x}+C$
②$\int \frac{1}{x^2}dx=-\frac{1}{x}+C$