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Table of Integrals

Basic Integrals | Rational Integrals | Trigonometric Integrals

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

2 .-) $\displaystyle \int \frac{dx}{x}=\ln \vert x\vert+C$

3 .-) $\displaystyle \int a^x \, dx=\frac{a^x}{\ln a}+C$

4 .-) $\displaystyle \int \tan x\, dx = \ln\vert\sec x\vert+C$

5 .- ) $\displaystyle \int \cot x\, dx = \ln\vert\sin x\vert+C$

6 .- ) $\displaystyle \int \sec x\, dx = \ln \vert\sec+\tan x\vert+C$

7 .- ) $\displaystyle \int \csc x\, dx = \ln \vert\csc-\cot x\vert+C$

8 .- ) $\displaystyle \int \frac{dx}{x^2-a^2}=\frac{1}{2a} \ln \left\vert \frac{x-a}{x+a}\right\vert+C, \qquad\mathrm{ if~~ } x^2>a^2$

9 .- ) $\displaystyle \int \frac{dx}{x^2-a^2}=\frac{1}{2a} \ln \left\vert \frac{a-x}{x+a}\right\vert+C, \qquad\mathrm{ if~~ } a^2>x^2$

10 .- ) $\displaystyle \int \frac{dx}{x^2+a^2}=\frac{1}{a}\arctan\left(\frac{x}{a}\right)+C$

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

12 .- ) $\displaystyle \int \frac{dx}{x^2 \pm a^2 }= \ln \left\vert x+\sqrt{x^2\pm a^2}\right\vert+C$

13 .- ) $\displaystyle \int \frac{dx}{x(a+bx)}=\frac{1}{a} \ln \left\vert \frac{x}{a+bx} \right\vert+C$

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

15 .- ) $\displaystyle \int \frac{dx}{x(a+bx)^2}= \frac{1}{a+bx} - \frac{1}{a^2}\ln \left\vert \frac{a+bx}{x} \right\vert+C$

Rational Integrals

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16 .- ) $\displaystyle \int x \sqrt{a+bx}\, dx = \frac{2(3bx-2a)\sqrt{(a+bx)^3} }{15b^2}+C$

17 .- ) $\displaystyle \int \frac{x\, dx}{\sqrt{a+bx} }=\frac{ 2(bx-2a)\sqrt{a+bx} }{3b^2}+C$

18 .- ) $\displaystyle \int x^2 \sqrt{a+bx}\, dx = \frac{2(15b^2x^2-12abx+8a^2)\sqrt{(a+bx)^3} }{105b^3}+C$

19 .- ) $\displaystyle \int \frac{x^2\, dx}{\sqrt{a+bx} }=\frac{ 2(3b^2x^2-4abx+8a^2)\sqrt{a+bx} }{15b^3}+C$

20 .- ) $\displaystyle \int \frac{dx}{x\sqrt{a+bx}}=\frac{1}{\sqrt a} \ln\left( \frac{\sqrt{a+bx}-\sqrt a}{\sqrt{a+bx}+\sqrt a}\right) + C \qquad\mathrm{ if~~ } a>0$

21 .- ) $\displaystyle \int \frac{dx}{x\sqrt{a+bx}}=\frac{2}{\sqrt{-a}} \arctan \sqrt{\frac{a+bx}{-a}} + C\qquad\mathrm{ if~~ } a<0$

22 .- ) $\displaystyle \int \frac{a+bx}{x}\, dx = 2\sqrt{a+bx}+a\int \frac{dx}{x\sqrt{a+bx}} + C$

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

24 .- ) $\displaystyle \int \sqrt{a^2-x^2}\, dx = \frac{1}{2}\left(x\sqrt{a^2-x^2}+a^2\arcsin\left(\frac{x}{a}\right)\right) + C$

25 .- ) $\displaystyle \int x\sqrt{a^2-x^2}\, dx = -\frac{1}{3}(a^2-x^2)^{3/2} + C$

26 .- ) $\displaystyle \int x^2 \sqrt{a^2-x^2}\, dx = \frac{x}{8}(2x^2-a^2)\sqrt{a^2-x^2}+\frac{a^4}{8}\arcsin\left(\frac{x}{a}\right) + C$

27 .- ) $\displaystyle \int \frac{x\, dx}{a^2-x^2} = -\sqrt{a^2-x^2} + C$

28 .- ) $\displaystyle \int \frac{x^2\, dx}{a^2-x^2} = -\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\arcsin\left(\frac{x}{a}\right) + C$

29 .- ) $\displaystyle \int (a^2-x^2)^{3/2}\, dx = \frac{x}{8}(5a^2-2x^2)\sqrt{a^2-x^2}+\frac{3a^4}{8}\arcsin\left(\frac{x}{a}\right)+ C$

30 .- ) $\displaystyle \int \frac{dx}{(a^2-x^2)^{3/2}} = \frac{x}{a^2\sqrt{a^2-x^2}} + C$

31 .- ) $\displaystyle \int \frac{x\, dx}{(a^2-x^2)^{3/2}} = \frac{1}{\sqrt{a^2-x^2}} + C$

32 .- ) $\displaystyle \int \frac{x^2\, dx}{(a^2-x^2)^{3/2}} = \frac{x}{\sqrt{a^2-x^2}}-\arcsin\left( \frac{x}{a}\right) + C$

33 .- ) $\displaystyle \int \frac{dx}{ x\sqrt{a^2-x^2} } = \frac{1}{a}\ln \left\vert \frac{a-\sqrt{a^2-x^2}}{x} \right\vert + C$

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

35 .- ) $\displaystyle \int \frac{dx}{x^3\sqrt{a^2-x^2}}=-\frac{\sqrt{a^2-x^2}}{2a^2x^2}+\frac{1}{2a^3}\ln\left\vert \frac{a-\sqrt{a^2-x^2}}{x} \right\vert + C$

36 .- ) $\displaystyle \int \frac{\sqrt{a^2-x^2}}{x}\, dx = \sqrt{a^2-x^2}-a\ln \left\vert \frac{a+\sqrt{a^2-x^2}}{x} \right\vert + C$

37 .- ) $\displaystyle \int \frac{\sqrt{a^2-x^2}}{x^2}\, dx =-\frac{\sqrt{a^2-x^2}}{x}-\arcsin\left(\frac{x}{a}\right) + C$

38 .- ) $\displaystyle \int \sqrt{x^2 \pm a^2}\, dx = \frac{1}{2}\left(x\sqrt{x^2 \pm a^2} \pm a^2\ln \vert x+\sqrt{x^2 \pm a^2} \vert\right) + C$

39 .- ) $\displaystyle \int x\sqrt{x^2 \pm a^2}\, dx = \frac{1}{3} (x^2\pm a^2)^{3/2} + C$

40 .- ) $\displaystyle \int x^2\sqrt{x^2 \pm a^2}\, dx = \frac{x}{8} (2x^2\pm a^2)\sqrt{x^2\pm a^2} -\frac{a^4}{8}\ln \vert x+\sqrt{x^2 \pm a^2} \vert + C$

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

42 .- ) $\displaystyle \int \frac{x^2 \, dx}{\sqrt{x^2\pm a^2}} = \frac{1}{2} \left(x\sqrt{x^2 \pm a^2} \mp a^2\ln \vert x+\sqrt{x^2 \pm a^2} \vert\right)+ C$

43 .- ) $\displaystyle \int (x^2 \pm a^2)^{3/2} \, dx = \frac{x}{8}(2x^2\pm 5a^2)\sqrt{x^2 \pm a^2} +\frac{3a^4}{8} \ln \vert x+\sqrt{x^2 \pm a^2} \vert + C$

44 .- ) $\displaystyle \int \frac{dx}{(x^2\pm a^2)^{3/2}} = \frac{\pm x}{a^2 \sqrt{x^2 \pm a^2}} + C$

45 .- ) $\displaystyle \int \frac{x\, dx}{(x^2\pm a^2)^{3/2}} = \frac{-1}{ \sqrt{x^2 \pm a^2}} + C$

46 .- ) $\displaystyle \int \frac{x^2\, dx}{(x^2\pm a^2)^{3/2}} = \frac{-x}{ \sqrt{x^2 \pm a^2}} + \ln \vert x+\sqrt{x^2 \pm a^2} \vert C$

47 .- ) $\displaystyle \int \frac{dx}{ x^2\sqrt{x^2 \pm a^2} } = \mp \frac{\sqrt{x^2 \pm a^2}}{a^2 x} + C$

48 .- ) $\displaystyle \int \frac{dx}{x^3 \sqrt{x^2 - a^2}} = \frac{ \sqrt{x^2-a^2}}{2a^2x^2}+\frac{1}{2a^3}\arccos\left( \frac{a}{x} \right) + C$

49 .- ) $\displaystyle \int \frac{\sqrt{x^2-a^2}}{x} \, dx = \sqrt{x^2-a^2}-a\arccos\left( \frac{a}{x} \right) + C$

50 .- ) $\displaystyle \int \frac{\sqrt{x^2\pm a^2}}{x^2}\, dx = -\frac{\sqrt{x^2 \pm a^2}}{x}+\ln \vert x+\sqrt{x^2 \pm a^2} \vert + C$

51 .- ) $\displaystyle \int \frac{dx}{x\sqrt{x^2+a^2}} = \frac{1}{a} \ln \left\vert\frac{x}{a+\sqrt{x^2+a^2}} \right\vert + C$

52 .- ) $\displaystyle \int \frac{dx}{x\sqrt{x^2-a^2}} = \frac{1}{a}\arccos\left( \frac{a}{x} \right) + C$

53 .- ) $\displaystyle \int \frac{dx}{x^3\sqrt{x^2+a^2}}=-\frac{\sqrt{x^2+a^2}}{2a^2x^2}+\frac{1}{2a^3}\ln \left\vert\frac{a+\sqrt{x^2+a^2}}{x} \right\vert + C$

54 .- ) $\displaystyle \int \frac{x^2+a^2}{x}\, dx = \sqrt{x^2+a^2}-a\ln \left\vert \frac{a+\sqrt{x^2+a^2}}{x}\right\vert + C$

55 .- ) $\displaystyle \int \sqrt{2ax-x^2}\, dx = \frac{x-a}{2}\sqrt{2ax-x^2}+\frac{a^2}{2}\arcsin \left(\frac{x-a}{a}\right) + C$

56 .- ) $\displaystyle \int \frac{dx}{\sqrt{2ax-x^2}} = 2\arcsin\sqrt{\frac{x}{2a}} + C=\arccos\left(1-\frac{x}{a} \right)+C$

57 .- ) $\displaystyle \int \frac{x^n \, dx}{\sqrt{2ax-x^2}} = -\frac{x^{n-1}\sqrt{2ax-x^2}}{n}+\frac{a(2n-1)}{n}\int \frac{ x^{n-1}\, dx}{\sqrt{2ax-x^2}} + C$

58 .- ) $\displaystyle \int x^n\sqrt{2ax-x^2}\, dx= \frac{\sqrt{2ax- x^2}}{a(1-2n)x^n}+\frac{n-1}{(2n-1)a}\int \frac{dx}{x^{n-1}\sqrt{2ax-x^2}}+ C$

59 .- ) $\displaystyle \int x^n\sqrt{2ax-x^2}\, dx = -\frac{x^{n-1}(2ax-x^2)^{3/2}}{n+2}+\frac{(2n+1)a}{n+2} \int x^{n-1}\sqrt{2ax-x^2}\, dx + C$

60 .- ) $\displaystyle \int \frac{ \sqrt{2ax-x^2}}{x^n} \, dx = \frac{(2ax-x^2)^{3/2}}{(3-2n)ax^n}+\frac{n-3}{(2n-3)a} + C$

61 .- ) $\displaystyle \int \frac{dx}{ \sqrt{2ax-x^2}^{3/2} } = \frac{x-a}{ a^2 \sqrt{2ax-x^2} } + C$

62 .- ) $\displaystyle \int \frac{dx}{\sqrt{2ax+x^2}}=\ln \vert x+a+\sqrt{2ax+x^2}\vert + C$

63 .- ) $\displaystyle \int \frac{dx}{a+bx+cx^2}=\frac{2}{4ac-b^2}\arctan\left( \frac{2cx+b}{\sqrt{4ac-b^2}}\right) + C$

64 .- ) $\displaystyle \int \frac{dx}{a+bx-cx^2}=\frac{1}{\sqrt c}\,\arcsin \left( \frac{2cx-b}{\sqrt{b+4ac}} \right) + C$

65 .- ) $\displaystyle \int \frac{dx}{a+bx-cx^2}=\frac{1}{b^2+4ac}\, \ln \left\vert \frac{ \sqrt{b^2+4ac}-b+2cx}{\sqrt{b^2+4ac}+b-2cx} \right\vert + C$

66 .- ) $\displaystyle \int \frac{dx}{\sqrt{a+bx+cx^2}} = \frac{1}{\sqrt{c}}\ln \vert 2cx+b+2\sqrt{c}\sqrt{a+bx+cx^2} \vert + C$

67 .- ) $\displaystyle \int \sqrt{a+bx+cx^2}\, dx = \frac{2cx+b}{4c}\sqrt{a+bx+cx^2}$

$\displaystyle -\frac{b^2-4ac}{8x^{3/2}} \ln \vert 2cx+b+2\sqrt{c}\sqrt{a+bx+cx^2} \vert + C$

68 .- ) $\displaystyle \int \sqrt{a+bx-cx^2}\, dx = \frac{2xc-b}{4c} \sqrt{a+bx-cx^2}+\frac{b^2+4ac}{8c^{3/2}} \arcsin \left( \frac{2cx-b}{\sqrt{b^2+4ac}} \right)+ C$

69 .- ) $\displaystyle \int \frac{x\, dx}{\sqrt{a+bx-cx^2}} = -\frac{\sqrt{a+bx-cx^2}}{c}+\frac{b}{2c^{3/2}}\,\arcsin \left(\frac{2cx-b}{\sqrt{b^2+4ac}}\right) + C$

70 .- ) $\displaystyle \int \frac{x\, dx}{\sqrt{a+bx+cx^2}} = \frac{\sqrt{a+bx+cx^2}}{c}- \frac{b}{2c^{3/2}} \ln \vert 2cx+b+2\sqrt{c}\sqrt{a+bx+cx^2} \vert + C$

Trigonometric Integrals

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71 .- ) $\displaystyle \int \sin^2 (ax) \, dx = \frac{1}{2a}(ax-\sin (ax)\cos(ax)) + C$

72 .- ) $\displaystyle \int \cos^2(ax)\, dx = \frac{1}{2a}(ax+\sin (ax)\cos (ax)) + C$

73 .- ) $\displaystyle \int \sin^n x\, dx = -\frac{\sin^{n-1}x \cos x}{n}+\frac{n-1}{n}\int \sin^{n-2} x\, dx + C$

74 .- ) $\displaystyle \int \cos^n x\, dx = \frac{\cos^{n-1} x \sin x}{n}+\frac{n-1}{n} \int \cos^{n-2} x\, dx+ C$

75 .- ) $\displaystyle \int \tan^n x\, dx =\frac{\tan^{n-1} x}{n-1}-\int \tan{n-2}x\, dx + C$

76 .- ) $\displaystyle \int \cot^n x\, dx =\frac{\cot^{n-1} x}{n-1}-\int \cot^{n-2}x\, dx + C$

77 .- ) $\displaystyle \int \sec^2 x\, dx = \tan x + C$

78 .- ) $\displaystyle \int \csc^2 x\, dx = -\cot x + C$

79 .- ) $\displaystyle \int \sec^n x\, dx = \frac{\tan x \sec^{n-2} x}{n-1}+\frac{n-2}{n-1} \int \sec^{n-2} x\, dx + C$

80 .- ) $\displaystyle \int \csc^n x \, dx = -\frac{\cot x \csc^{n-2} x}{n-1} +\frac{n-2}{n-1} \int \csc^{n-2} x\, dx + C$

81 .- ) $\displaystyle \int \sec x \tan x \, dx = \sec x + C$

82 .- ) $\displaystyle \int \csc x \cot x = -\csc x + C$

83 .- ) $\displaystyle \int \cos^n x \sin^m x\, dx = + C$

$\displaystyle = \frac{ \cos^{m-1} x \sin^{n+1} x }{m+n} + \frac{m-1}{m+n} \int \cos^{m-2} x \sin^n x \, dx$

$\displaystyle = \frac{ \sin^{n-1} x \cos^{m+1} x }{m+n} + \frac{n-1}{m+n} \int \cos^{m} x \sin^{n-2} x \, dx$

$\displaystyle = \frac{ \cos^{m+1} x \sin^{n+1} x }{m+1} + \frac{m+n+2}{m+1} \int \cos^{m+2} x \sin^n x \, dx$

$\displaystyle = \frac{ \cos^{m+1} x \sin^{n+1} x }{n+1} + \frac{m+n+2}{n+1} \int \cos^{m} x \sin^{n+2} x \, dx$

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