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附录积分表大全（上）

1.$\int{{{x}^{n}}dx=\frac{{{x}^{n+1}}}{n+1}}+C,(n\ne 1)$

2.$\int{kdx=kx+C}$

3.$\int{\frac{dx}{x}=\ln }\left| x \right|+C$

4.$\int{{{e}^{x}}dx={{e}^{x}}+C}$

5.$\int{{{a}^{x}}dx=\frac{{{a}^{x}}}{\ln a}+C},(a>0,a\ne 1)$

6.$\int{\sin xdx=-\cos x+C}$

7.$\int{\cos xdx=\sin x+C}$

8.$\int{{{\sec }^{2}}xdx=\tan x+C}$

9.$\int{{{\csc }^{2}}xdx=-\cot x+C}$

10.$\int{\sec x\tan xdx=\sec x+C}$

11.$\int{\csc x\cot xdx=-\csc x+C}$

12.$\int{\tan xdx=\ln \left| \sec x \right|}+C$

13.$\int{\cot xdx=\ln \left| \sin x \right|}+C$

14.$\int{\sinh xdx=\cosh x}+C$

15.$\int{\cosh xdx=\sinh x}+C$

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

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

18.$\int{\frac{dx}{x\sqrt{{{x}^{2}}-{{a}^{2}}}}}=\frac{1}{a}{{\sec }^{-1}}\left| \frac{x}{a} \right|+C$

19.$\int{\frac{dx}{\sqrt{{{a}^{2}}+{{x}^{2}}}}}={{\sinh }^{-1}}\frac{x}{a}+C$

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

21.$\int{{{(ax+b)}^{n}}dx=\frac{{{(ax+b)}^{n+1}}}{a(n+1)}}+C,n\ne -1$

22.$\int{x{{(ax+b)}^{n}}dx=\frac{{{(ax+b)}^{n+1}}}{{{a}^{2}}}}\left[ \frac{ax+b}{n+2}-\frac{b}{n+1} 23.\right]+C,n\ne -1,-2$

24.$\int{{{(ax+b)}^{-1}}dx=\frac{1}{a}}\ln \left| ax+b \right|+C$

25.$\int{x{{(ax+b)}^{-1}}dx=\frac{x}{a}}-\frac{b}{{{a}^{2}}}\ln \left| ax+b \right|+C$

26.$\int{x{{(ax+b)}^{-2}}dx=\frac{1}{{{a}^{2}}}}\left[ \ln \left| ax+b \right|+\frac{b}{ax+b} \right]+C$

27.$\int{\frac{dx}{x(ax+b)}=\frac{1}{b}}\left[ \ln \frac{x}{ax+b} \right]+C$

28.$\int{{{(\sqrt{ax+b})}^{n}}dx=\frac{2}{a}}\frac{({{\sqrt{ax+b)}}^{n+2}}}{n+2}+C,n\ne -2$

$\int{\frac{\sqrt{ax+b}}{x}dx=2}\sqrt{ax+b}+b\int{-\frac{dx}{x\sqrt{ax+b}}}$

29.(a)$\int{\frac{dx}{x\sqrt{ax+b}}=\frac{1}{\sqrt{b}}\ln }\left| \frac{\sqrt{ax+b}-\sqrt{b}}{\sqrt{ax+b}+\sqrt{b}} \right|+C$

(b)$\int{\frac{dx}{x\sqrt{ax-b}}=\frac{2}{\sqrt{b}}{{\tan }^{-1}}}\frac{\sqrt{ax-b}}{b}+C$

30.$\int{\frac{\sqrt{ax+b}}{{{x}^{2}}}}dx=-\frac{\sqrt{ax+b}}{x}+\frac{a}{2}\int{\frac{dx}{x\sqrt{ax+b}}+C}$

31.

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

32.$\int{\frac{dx}{{{a}^{2}}+{{x}^{2}}}=\frac{1}{a}}{{\tan }^{-1}}\frac{x}{a}+C$

33.$\int{\frac{dx}{{{({{a}^{2}}+{{x}^{2}})}^{2}}}=\frac{x}{2{{a}^{2}}({{a}^{2}}+{{x}^{2}})}}+\frac{1}{2{{a}^{3}}}{{\tan }^{-1}}\frac{x}{a}+C$

34.$\int{\frac{dx}{\sqrt{{{a}^{2}}+{{x}^{2}}}}={{\sinh }^{-1}}\frac{x}{a}}+C=\ln \left( x+\sqrt{{{a}^{2}}+{{x}^{2}}} \right)+C$

35.$\int{\sqrt{{{a}^{2}}+{{x}^{2}}}}dx=\frac{x}{2}\sqrt{{{a}^{2}}+{{x}^{2}}}+\frac{{{a}^{2}}}{2}\ln x\left( +\sqrt{{{a}^{2}}+{{x}^{2}}} \right)+C$

36.$\int{{{x}^{2}}\sqrt{{{a}^{2}}+{{x}^{2}}}}dx=\frac{x}{8}\left( {{a}^{2}}+2{{x}^{2}} \right)\sqrt{{{a}^{2}}+{{x}^{2}}}-\frac{{{a}^{4}}}{8}\ln \left( x+\sqrt{{{a}^{2}}+{{x}^{2}}} \right)+C$

37.$\int{\frac{\sqrt{{{a}^{2}}+{{x}^{2}}}}{x}}dx=\sqrt{{{a}^{2}}+{{x}^{2}}}-a\ln \left| \frac{a+\sqrt{{{a}^{2}}+{{x}^{2}}}}{x} \right|+C$

38.$\int{\frac{\sqrt{{{a}^{2}}+{{x}^{2}}}}{{{x}^{2}}}}dx=\ln \left( x+\sqrt{{{a}^{2}}+{{x}^{2}}} \right)-\frac{\sqrt{{{a}^{2}}+{{x}^{2}}}}{x}+C$

39.$\int{\frac{{{x}^{2}}}{\sqrt{{{a}^{2}}+{{x}^{2}}}}}dx=-\frac{{{a}^{2}}}{2}\ln \left( x+\sqrt{{{a}^{2}}+{{x}^{2}}} \right)+\frac{x\sqrt{{{a}^{2}}+{{x}^{2}}}}{2}+C$

40.$\int{\frac{dx}{x\sqrt{{{a}^{2}}+{{x}^{2}}}}}=-\frac{1}{a}\ln \left| \frac{a+\sqrt{{{a}^{2}}+{{x}^{2}}}}{x} \right|+C$

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

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

43. $\int{\frac{dx}{{{\left( {{a}^{2}}-{{x}^{2}} \right)}^{2}}}}=\frac{x}{2{{a}^{2}}({{a}^{2}}-{{x}^{2}})}+\frac{1}{4{{a}^{3}}}\ln \left| \frac{x+a}{x-a} \right|+C$

44. $\int{\frac{dx}{\sqrt{{{a}^{2}}-{{x}^{2}}}}}={{\sin }^{-1}}\frac{x}{a}+C$

45. $\int{\sqrt{{{a}^{2}}-{{x}^{2}}}}dx=\frac{x}{2}\sqrt{{{a}^{2}}-{{x}^{2}}}+\frac{{{a}^{2}}}{2}{{\sin }^{-1}}\frac{x}{a}+C$

46. $\int{{{x}^{2}}\sqrt{{{a}^{2}}-{{x}^{2}}}}dx=\frac{{{a}^{4}}}{8}{{\sin }^{-1}}\frac{x}{a}-\frac{1}{8}x\sqrt{{{a}^{2}}-{{x}^{2}}}\left( {{a}^{2}}-2{{x}^{2}} \right)+C$

47. $\int{\frac{\sqrt{{{a}^{2}}-{{x}^{2}}}}{x}}dx=\sqrt{{{a}^{2}}-{{x}^{2}}}-a\ln \left| \frac{a+\sqrt{{{a}^{2}}-{{x}^{2}}}}{x} \right|+C$

48. $\int{\frac{\sqrt{{{a}^{2}}-{{x}^{2}}}}{{{x}^{2}}}}dx=-{{\sin }^{-1}}\frac{x}{a}-\frac{\sqrt{{{a}^{2}}-{{x}^{2}}}}{x}+C$

49. $\int{\frac{{{x}^{2}}}{\sqrt{{{a}^{2}}-{{x}^{2}}}}}dx=\frac{{{a}^{2}}}{2}{{\sin }^{-1}}\frac{x}{a}-\frac{1}{2}x\sqrt{{{a}^{2}}-{{x}^{2}}}+C$

50. $\int{\frac{dx}{x\sqrt{{{a}^{2}}-{{x}^{2}}}}}=-\frac{1}{a}\ln \left| \frac{a+\sqrt{{{a}^{2}}-{{x}^{2}}}}{x} \right|+C$

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

52. $\int{\frac{dx}{\sqrt{{{x}^{2}}-{{a}^{2}}}}}=\ln \left| x+\sqrt{{{x}^{2}}-{{a}^{2}}} \right|+C$

53. $\int{\sqrt{{{x}^{2}}-{{a}^{2}}}dx}=\frac{x}{2}\sqrt{{{x}^{2}}-{{a}^{2}}}-\frac{{{a}^{2}}}{2}\ln \left| x+\sqrt{{{x}^{2}}-{{a}^{2}}} \right|+C$

54. $\int{{{\left( \sqrt{{{x}^{2}}-{{a}^{2}}} \right)}^{n}}dx}=\frac{x{{\left( \sqrt{{{x}^{2}}-{{a}^{2}}} \right)}^{n}}}{n+1}-\frac{n{{a}^{2}}}{n+1}\int{{{\left( \sqrt{{{x}^{2}}-{{a}^{2}}} \right)}^{n-2}}dx,(n\ne -1)}$

55. $\int{\frac{dx}{{{\left( \sqrt{{{x}^{2}}-{{a}^{2}}} \right)}^{n}}}}=\frac{x{{\left( \sqrt{{{x}^{2}}-{{a}^{2}}} \r

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附录 积分表大全（上）

附录积分表大全（上）