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素材/整合素材/高数素材/常见积分表.md

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+常用不定积分表
+一、基本初等函数积分
+ $\int 0 \, dx =C$
+ $\int k \, dx = kx + C$ （ k  为常数）
+ $\int x^\mu \, dx = \frac{x^{\mu+1}}{\mu+1} + C$ （ $\mu \neq -1$ ）
+ $\int \frac{1}{x} \, dx = \ln|x| + C$ 
+$\int a^x \, dx = \frac{a^x}{\ln a} + C$ （ $a>0,a\neq1$ ）
+ $\int e^x \, dx = e^x + C$ 
+二、三角函数积分
+ $\int \sin x \, dx = -\cos x + C$ 
+ $\int \cos x \, dx = \sin x + C$ 
+ $\int \tan x \, dx = -\ln|\cos x| + C$
+ $\int \cot x \, dx = \ln|\sin x| + C$
+ $\int \sec x \, dx = \ln|\sec x + \tan x| + C$ 
+ $\int \csc x \, dx = \ln|\csc x - \cot x| + C$ 
+ $\int \sec^2 x \, dx = \tan x + C$ 
+ $\int \csc^2 x \, dx = -\cot x + C$ 
+ $\int \sec x \tan x \, dx = \sec x + C$ 
+ $\int \csc x \cot x \, dx = -\csc x + C$ 
+ $\int \sin^2 x \, dx = \frac{x}{2} - \frac{\sin 2x}{4} + C$ 
+ $\int \cos^2 x \, dx = \frac{x}{2} + \frac{\sin 2x}{4} + C$ 
+三、反三角函数积分
+ $\int \arcsin x \, dx = x\arcsin x + \sqrt{1-x^2} + C$ 
+ $\int \arccos x \, dx = x\arccos x - \sqrt{1-x^2} + C$ 
+ $\int \arctan x \, dx = x\arctan x - \frac{1}{2}\ln(1+x^2) + C$ 
+ $\int \text{arccot } x \, dx = x\text{arccot } x + \frac{1}{2}\ln(1+x^2) + C$ 
+四、含根式的积分（ a>0 ）
+ $\int \frac{1}{\sqrt{a^2 - x^2}} \, dx = \arcsin\frac{x}{a} + C$ 
+ $\int \frac{1}{\sqrt{x^2 + a^2}} \, dx = \ln\left(x + \sqrt{x^2 + a^2}\right) + C$ 
+ $\int \frac{1}{\sqrt{x^2 - a^2}} \, dx = \ln\left|x + \sqrt{x^2 - a^2}\right| + C$ 
+ $\int \sqrt{a^2 - x^2} \, dx = \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}\arcsin\frac{x}{a} + C$ 
+ $\int \sqrt{x^2 + a^2} \, dx = \frac{x}{2}\sqrt{x^2 + a^2} + \frac{a^2}{2}\ln\left(x + \sqrt{x^2 + a^2}\right) + C$ 
+ $\int \sqrt{x^2 - a^2} \, dx = \frac{x}{2}\sqrt{x^2 - a^2} - \frac{a^2}{2}\ln\left|x + \sqrt{x^2 - a^2}\right| + C$ 
+五、含分式的积分（ $a\neq0$ ）
+ $\int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a}\arctan\frac{x}{a} + C$ 
+ $\int \frac{1}{x^2 - a^2} \, dx = \frac{1}{2a}\ln\left|\frac{x - a}{x + a}\right| + C$ 
+ $\int \frac{1}{ax + b} \, dx = \frac{1}{a}\ln|ax + b| + C$ 
+六、指数与对数结合积分
+ $\int x e^x \, dx = (x-1)e^x + C$ 
+$\int x \ln x \, dx = \frac{x^2}{2}\ln x - \frac{x^2}{4} + C$ 
+ $\int e^x \sin x \, dx = \frac{e^x}{2}(\sin x - \cos x) + C$ 
+ $\int e^x \cos x \, dx = \frac{e^x}{2}(\sin x + \cos x) + C$ 
+七、常用凑微分积分
+ $\int \frac{1}{\sqrt{1 - x^2}} \, dx = \arcsin x + C = -\arccos x + C$ 
+ $\int \frac{1}{1 + x^2} \, dx = \arctan x + C = -\text{arccot } x + C$ 
+ $\int \frac{1}{x\ln x} \, dx = \ln|\ln x| + C$ 
+ $\int \frac{e^{\sqrt{x}}}{\sqrt{x}} \, dx = 2e^{\sqrt{x}} + C$ 
+ $\int \frac{\sin\sqrt{x}}{\sqrt{x}} \, dx = -2\cos\sqrt{x} + C$