基本积分公式

$$1. \ \int{x^k}dx \ = \ \frac{1}{k \ + \ 1}x^{k + 1} \ + \ C,k \ne -1;$$

$
\left\{
\begin{matrix}
\int{\frac{1}{x^2}}dx = & -\frac{1}{x} + C \\
\int{\frac{1}{\sqrt{x}}}dx = & 2\sqrt{x} + C \\
\end{matrix}
\right.
$

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$$2. \ \int{\frac{1}{x}}dx \ = \ \ln|x| \ + \ C$$

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$$3. \ \int{e^x}dx \ = \ e^x \ + \ C;\int{a^x}dx \ = \ \frac{a^x}{\ln{a}} \ + \ C,a \ > 0且a \ \ne \ 1$$

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$$4. \ \int{\sin{x}}dx \ = \ -\cos{x} \ + \ C$$

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$$5. \ \int{\cos{x}}dx \ = \ \sin{x} \ + \ C$$

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$$6. \ \int\tan{x}dx \ = \ -\ln|\cos{x}| \ + \ C$$

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$$7. \ \int{\cot{x}}dx \ = \ \ln|\sin{x}| \ + \ C$$

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$$8. \ \int\frac{dx}{\cos{x}} \ = \ \int{\sec{x}dx} \ = \ \ln{|\sec{x} \ + \ \tan{x}| \ + \ C}$$

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$$9. \ \int\frac{dx}{\sin{x}} \ = \ \int{\csc{x}}dx \ = \ \ln|\csc{x} \ - \ \cot{x}| \ + \ C$$

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$$10. \ \int\sec{^2x}dx \ = \ \tan{x} \ + \ C$$

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$$11. \ \int\csc{^2x}dx \ = \ -\cot{x} \ + \ C$$

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$$12. \ \int{\sec{x} \tan{x}}dx \ = \ \sec{x} \ + \ C$$

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$$13. \ \int{\csc{x} \cot{x}}dx \ = \ -\csc{x} \ + \ C$$

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$$14. \ \int\frac{1}{1 \ + \ x^2}dx \ = \ \arctan{x} \ + \ C$$

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$$15. \ \int\frac{1}{a^2 \ + \ x^2}dx \ = \ \frac{1}{a}\arctan{\frac{x}{a}} \ + \ C(a \ > \ 0)$$

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$$16. \ \int\frac{1}{\sqrt{1 \ - \ x^2}} \ = \ \arcsin{x} \ + \ C$$

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$$17. \ \int\frac{1}{\sqrt{a^2 \ - \ x^2}} \ = \ \arcsin{\frac{x}{a}} \ + \ C$$

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$$18. \ \int\frac{1}{\sqrt{x^2 \ + \ a^2}}dx \ = \ \ln{(x \ + \ \sqrt{x^2 \ + \ a^2})} \ + \ C(常见 \ a=1)$$

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$$19. \ \int\frac{1}{\sqrt{x^2 \ - \ a^2}}dx \ = \ \ln{|x \ + \ \sqrt{x^2 \ - \ a^2}|} \ + \ C(|x| \ > \ |a|)$$

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

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

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$$22. \ \int\sqrt{a^2 \ - \ x^2}dx \ = \ \frac{a^2}{2}\arcsin{\frac{x}{a}} \ + \ \frac{x}{2}\sqrt{a^2 \ - \ x^2} \ + \ C(a \ > |x| \ \ge 0)$$

证明：

$求\int\sqrt{a^2 \ - \ x^2}dx(a \ > \ 0)$

$设x \ = \ a\sin{t}$

$则：dx \ = \ a\cos{t}dt,t \ = \ \arcsin{\frac{x}{a}},-a \ \le x \ \le a,-\frac{\Pi}{2} \ \le \ t \ \le \ \frac{\Pi}{2}$

$所以：$

$\int\sqrt{a^2 \ - \ x^2}dx$

$= \ \sqrt{a^2 \ - \ a^2\sin{^2t}} \ * \ a\cos{t}dt$

$= \ a^2\int\cos{^2t}dt$

$= \ \frac{a^2}{2}\int{(1 \ + \ \cos{2t})}dt$

$= \ \frac{a^2}{2}t \ + \ \frac{a^2}{4}\sin{2t} \ + \ C$

$= \ \frac{a^2}{2}t \ + \ \frac{a^2}{2}\sin{t}\cos{t} \ + \ C $

$= \ \frac{a^2}{2}\arcsin{\frac{x}{a}} \ + \ \frac{x}{2}\sqrt{a^2 \ - \ x^2} \ + \ C(这里的转化看设的x \ = \ a\sin{t}，\sin{t} \ = \ \frac{x}{a}, \ cos{t} \ = \ \frac{\sqrt{a^2 \ - \ x^2}}{a})$

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$$23. \ \int\sin{^2x}dx \ = \ \frac{x}{2} \ - \ \frac{\sin{2x}}{4} \ + \ C(由此式子转化而来:\sin{^2x} \ = \ \frac{1 \ - \ \cos{2x}}{2})$$

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$$24. \ \int\cos{^2x}dx \ = \ \frac{x}{2} \ + \ \frac{\sin{2x}}{4} \ + \ C(由此式子转化而来:\sin{^2x} \ = \ \frac{1 \ + \ \cos{2x}}{2})$$

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$$25. \ \int\tan{^2x} \ = \ \tan{x} \ - \ x \ + \ C(\tan{^2x} \ = \ \sec{^2x} \ - \ 1)$$

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$$26. \ \int\cot{^2x} \ = \ -\cot{x} \ - \ x \ + \ C(\cot{^2x} \ = \ \csc{^2x} \ - \ 1)$$

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GGB！！！