基本积分公式

基本求导公式

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

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$$\int \frac{1}{x}dx=\ln \mid x\mid$$ $$\int a^{x}dx=\frac{a^x}{\ln a}+C$$

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$$\int \tan xdx=-\ln \mid \cos \mid +C$$ $$\int \cot xdx=\ln \mid \sin x\mid +C$$

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$$\int \sec xdx=\ln \mid \sec x+\tan x\mid +C$$ $$\int \csc xdx=\ln \mid \csc x-\cot x\mid +C$$

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$$\int {\sec }^{2}xdx=\tan x+C$$ $$\int {\csc }^{2}xdx=-\cot x+C$$ $$\int \sec x\tan xdx=\sec x+C$$ $$\int \csc x\cot xdx=-\csc x+C$$

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

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$$\int \frac{1}{\sqrt{x^2+a^2}}dx=\ln (x+\sqrt{x^2+a^2})+C$$ $$\int \frac{1}{\sqrt{x^2-a^2}}dx=\ln \mid x+\sqrt{x^2-a^2}\mid +C$$

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

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

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$$\int {\sin }^{2}xdx=\int \frac{1-\cos 2x}{2}dx=\frac{1}{2}x-\frac{1}{4}\sin 2x+C$$ $$\int {\cos }^{2}xdx=\int \frac{1+\cos 2x}{2}dx=\frac{1}{2}+\frac{1}{4}\sin 2x+C$$ $$\int {\tan }^{2}xdx=\int {\sec }^{2}x-1dx=\tan x-x+C$$ $$\int {\cot }^{2}xdx=\int {\csc }^{2}x-1dx=-\cot x-x+C$$