L2 – Substitution

Identities Used in this Lecture:

$\int 1.\, dx=x+\, {\rm C}$

$\int x^{n} \, dx=\frac{x^{n\, +\, 1} }{n+1} +{\rm C}$

$\int (a\, x+b)^{n} \, dx=\frac{(a\, x+b)^{n\, +\, 1} }{a\, (n+1)} \, \, +{\rm C}$

$\int \sin x\, dx=-\cos x+{\rm C}$

$\int {\rm cosec}^{2} x\, dx=-\cot x+{\rm C}$

$\int \cos x\, dx=\sin x+{\rm C}$

$\int \sec x\tan x\, dx=\sec x+{\rm C}$

$\int \sin \, (a\, x+b)\, dx=\frac{-\cos \, (a\, x+b)}{a} +{\rm C}$

$\int \tan x\, dx=\log |\sec x|+{\rm C}=-\log |\cos x|+\, {\rm C}$

$\int \sec x\, dx=\log |\sec x+\tan x|+{\rm C}$

$\int \sec ^{2} x\, dx=\tan x+{\rm C}$

$\int {\rm cosec}\, x\, \cot x\, dx=-{\rm cosec}\, x+{\rm C}$

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

$\int {\frac{dx}{1+x^{2}} = \tan^{-1} x+{\rm C}} = -\cot^{-1} x+{\rm C}$

$\int {\frac{dx}{x\sqrt{x^{2} -1\, }} = \sec ^{-1} x+{\rm C}} = -\cosec^{-1} x+{\rm C}$

$\int e^{x} \, dx=e^{x} +{\rm C}$

$\int \frac{1}{x} dx=\log |x|+{\rm C}$

$\int \frac{1}{a\, x+b} dx=\frac{\log |a\, x+b|}{a} +{\rm C}$

$\int e^{a\, x+b} \, dx=\frac{e^{a\, x+b} }{a} +{\rm C}$

$\int \cot x\, dx=\log |\sin x|+{\rm C}$

$\int {\rm cosec}\, x\, dx=\log |{\rm cosec}\, x-\cot x|+{\rm C}$

PART - 1

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Logic for this lecture:

There are a lot of ways to perform substitution in integration.

First way is to substitute the function only if its derivative is also occurring in question. Sometimes, you need to simplify the function to know whether you can apply substitution or not.

Questions Discussed in this lecture:
1.
$\int _{}^{}2x.\sin (x^{2} +1)\, dx = -\cos (x^{2} +1)+C$

2. $\int _{}^{}\frac{\tan ^{4} \sqrt{x} \sec ^{2} \sqrt{x} }{\sqrt{x} } \, dx = \frac{2}{5} \tan ^{5} \sqrt{x} +C$

3. $\int _{}^{}\frac{\sin (\tan ^{-1} x)}{1+x^{2} } \, dx = -\cos (\tan ^{-1} x)+C$

4. $\int _{}^{}\frac{(\log x)^{2} }{x} dx = \frac{1}{3} (\log |x|)^{3} +C$

5. $\int _{}^{}\sin ^{3} x\cos ^{2} x\, dx = -\frac{1}{3} \cos ^{3} x+\frac{1}{5} \cos ^{5} x+C$

6. $\int \cos ^{5} x\, dx = \sin x\, -\, \frac{2}{3} \sin ^{3} x+\frac{1}{5} \sin ^{5} x+C$

7. $\int _{}^{}\cot x\log \sin x\, dx = \frac{1}{2} (\log \sin x)^{2} +C$

8. $\int _{}^{}\frac{\sin x}{1+\cos x} \, dx = -\log |1+\cos x|+C$

9. $\int \frac{1}{x\log x\log (\log x)} dx = \log |\log \, (\log x)|+C$

10. $\int \frac{e^{\sqrt{x} } }{\sqrt{x} } dx = 2e^{\sqrt{x} } +C$

PART - 2

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Logic for this lecture:

Same as in previous part…

Questions Discussed in this lecture:

11. $\int _{}^{}\sin ^{3} x\, dx = -\frac{3}{4} \cos x+\frac{1}{12} \cos 3x+C or -\cos x+\frac{1}{3} \cos ^{3} x+C$

12. $\int \frac{1}{x\, (2+3\log x)} \, dx = \frac{1}{3} \log |2+3\log x|+C$

13. $\int \frac{1-\sin 2x}{x+\cos ^{2} x} \, dx = \log |x+\cos ^{2} x|+C$

14. $\int \frac{3ax}{b^{2} +c^{2} x^{2} } dx = \frac{3a}{2c^{2} } \log |b^{2} +c^{2} x^{2} |+C$

15. $\int \frac{\sin 2x}{a^{2} \sin ^{2} x+b^{2} \cos ^{2} x} \, dx = \frac{1}{(a^{2} -b^{2} )} \log |a^{2} \sin ^{2} x+b^{2} \cos ^{2} x|+C$

16. $\int \tan ^{8} x\sec ^{4} x \, dx = \frac{\tan ^{11} x}{11} +\frac{\tan ^{9} x}{9} +C$

17. $\int \frac{x^{2} }{1+x^{3} } dx = \frac{1}{3} \log |1+x^{3} |+C$

18. $\int \frac{1-\cos x}{\sin x} \, dx=2\log \left|\, \sec \frac{x}{2} \, \right|+C$

19. $\int _{}^{}\frac{\sin ^{3} x+\cos ^{3} x}{\sin ^{2} x.\cos ^{2} x} \, dx = \sec x-{\rm cosec}\, x\, +C$

20. $\int _{}^{}\frac{dx}{\sin x\cos ^{3} x} = \log |\tan x|+\frac{1}{2} \tan ^{2} x+C$

PART - 3

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Logic for this lecture:

Same as in previous part…

Questions Discussed in this lecture:

21. $\int \frac{1+\tan x}{1-\tan x} \, dx = -\log |\cos x-\sin x|+C$

22. $\int _{}^{}\frac{\cos 2x}{(\sin x+\cos x)^{2} } \, dx = \log \left|\sin x+\cos x\right| +C$

23. $\int _{}^{}\frac{dx}{x+x\log x} = \log |1+\log x|+C$

24. $\int _{}^{}\frac{1}{1+\tan x} dx = \frac{x}{2} +\frac{1}{2} \log |\cos x+\sin x|+C$

25. $\int _{}^{}\frac{1}{x-\sqrt{x} } dx = 2\log |\sqrt{x} -1|+C$

26. $\int _{}^{}\frac{1}{\cos ^{2} x(1-\tan x)^{2} } dx = \frac{1}{1-\tan x} +C$

27. $\int _{}^{}\frac{2\cos x-3\sin x}{6\cos x+4\sin x} dx = \frac{1}{2} \log |2\sin x+3\cos x|+\, C$

28. $\int _{}^{}\frac{e^{\tan ^{-1} x} }{1+x^{2} } dx = e^{\tan ^{-1} x} +C$

29. $\int _{}^{}\frac{e^{2x} -1}{e^{2x} +1} \, dx = \log |e^{x} +e^{-x} |+\, C$

30. $\int _{}^{}\sqrt{\sin 2x} \cos 2x \, dx = \frac{1}{3} (\sin 2x)^{\frac{3}{2} } +C$

PART - 4

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Logic for this lecture:

Same as in previous part…

Questions Discussed in this lecture:

31. $\int _{}^{}\frac{\cos x}{\sqrt{1+\sin x} } dx = 2\sqrt{1+\sin x} +C$

32. $\int _{}^{}\frac{1}{1+\cot x} \, dx = \frac{x}{2} -\frac{1}{2} \log |\cos x+\sin x|+\, C$

33. $\int _{}^{}\frac{\sqrt{\tan x} }{\sin x\cos x} \, dx = 2\sqrt{\tan x} +C$

34. $\int _{}^{}\frac{(x+1)(x+\log x)^{2} }{x} \, dx = \frac{1}{3} (x+\log x)^{3} +C$

35. $\int _{}^{}\frac{x^{3} \sin (\tan ^{-1} x^{4} )}{1+x^{8} } \, dx = -\frac{1}{4} \cos (\tan ^{-1} x^{4} )+C$

36. $\int _{}^{}\frac{10x^{9} +10^{x} \log 10}{x^{10} +10^{x} } \, dx = \log |10^{x} +x^{10} |+\, C$

37. $\int _{}^{}\sin ^{3} x\cos ^{3} x\, dx = \frac{1}{6} \cos ^{6} x-\frac{1}{4} \cos ^{4} x+C$

38. $\int \frac{1+\cot x}{x+\log \sin x} \, dx = \log |x+\log \sin x|+C$

39. $\int \tan ^{2} x\, \, \sec ^{4} x\, dx = \frac{\tan ^{5} x}{5} +\frac{\tan ^{3} x}{3} +C$

40. $\int e^{\log \, (x+1)\, -\, \log x} dx = x+\log x+C$

PART - 5

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Logic for this lecture:

Same as in previous part…

Questions Discussed in this lecture:

41. $\int \frac{\sin x}{\sin 2x} \, \, dx = \frac{1}{2} \log |\sec x+\tan x|+C$

42. $\int _{}^{}\tan ^{4} x\, dx = \frac{1}{3} \tan ^{3} x-\tan x+x+C$

43. $\int \frac{2x+3}{x^{2} +3x} \, dx = \log |x^{2} +3x|+C$

44. $\int \frac{1+\cos x}{x+\sin x} \, dx = \log |x+\sin x|+C$

45. $\int _{}^{}\cos ^{4} 2x\, dx = \frac{3x}{8} +\frac{1}{8} \sin 4x+\frac{1}{64} \sin 8x+C$

46. $\int _{}^{}\frac{\sin ^{2} x}{1+\cos x} \, dx = x-\sin x+C$

47. $\int \frac{\sin ^{6} x}{\cos ^{8} x} \, dx = \frac{\tan ^{7} x}{7} +C$

48. $\int _{}^{}\frac{\cos x-\sin x}{1+\sin 2x} \, dx = -\frac{1}{\cos x+\sin x} \, +C$

49. $\int _{}^{}\tan ^{3} 2x\sec 2x \, dx = \frac{1}{6} \sec ^{3} 2x-\frac{1}{2} \sec 2x+C$

50. $\int \frac{dx}{\cos ^{2} x+\sin 2x} = \frac{1}{2} \log |1+2\tan x|+C$

PART - 6

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Logic for this lecture:

Same as in previous part…

Questions Discussed in this lecture:

$$\int \tan x dx$$

$$\int \cot x dx$$

$$\int \sec x dx$$

$$\int \cosec x dx$$

PART - 7

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Logic for this lecture:

Same as in previous part…

Questions Discussed in this lecture:

51. $\int\frac{cos{2}x}{(cos{x}+sin{x})^2}dx = \cos{x}+sin{x}|+C$

52. $\int\frac{dx}{sin{x}{cos}^3{x}} = {|}tan{x}|+\frac{1}{2}{tan}^2{x}+C$

53. $\int\frac{x^2+4x}{x^3+6x^2+5}dx = \frac{1}{3}log{|}x^3+6x^2+5|+C$

54. $\int{sec{x}.log{(}sec{x}+tan{x})dx}\ =\ \frac{(log{|}sec{x}+tan{x}|)^2}{2}+C$

55. $\int{cot{x}.logsin{x}dx} =\frac{(logsin{x})^2}{2}+C$

56. $\int{{cot}^3{x}{cosec}^4{x}dx} = -\left(\frac{{cot}^6{x}}{6}+\frac{{cot}^4{x}}{4}\right)+C$

57. $\int\frac{x^{e-1}+e^{x-1}}{x^e+e^x}dx = \frac{1}{e}log{|}x^e+e^x|+C$

58. $\int{\frac{e^x(1+x)}{{cos}^2{(}e^xx)}dx = {\tan (}xe^x)+C}$

PART - 8

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Logic for this lecture:

Same as in previous part…

Questions Discussed in this lecture:

59. $\int{\frac{{cos}^5{x}}{sin{x}}dx} = log{|}sin{x}|+\frac{1}{4}{sin}^4{x}-{sin}^2{x}+C$

60. $\int\frac{4x+1}{\sqrt{2x^2+x-3}}dx=2\sqrt{2x^2+x-3}+C$

61. $\int{\frac{dx}{e^x+e^{-x}}={tan}^{-1}{(}e^x)+C}$

62. $\int{cos{6}x\sqrt{1+sin{6}x}dx=\frac{1}{9}(1+sin{6}x)^\frac{3}{2}+C}$

63. $\int{{cos}^2{x}.e^{logsin{x}}dx=-\frac{1}{4}{cos}^4{x}}+C$

64. $\int{\frac{\sqrt{tan{x}}}{sin{2}x}dx}=2\sqrt{tan{x}}+C$

65. $\int\frac{{cos}^9{x}}{sin{x}}dx=log{|}sin{x}|-2{sin}^2{x}+\frac{3}{2}{sin}^4{x}-\frac{2}{3}{sin}^6{x}+\frac{1}{8}{sin}^8{x}+C$

66. $\int{e^{3log{x}}(x^4+1)^{-1}dx=\frac{1}{4}log{(}x^4+1)+C}$

67. $\int_{0}^{\frac{\pi}{4}}{{sin}^3{2}tcos{2}tdt=\frac{1}{8}}$

69. $\int_{4}^{9}{\frac{\sqrt x}{(30-x^\frac{3}{2})^2}dx}=\frac{19}{99}$

PART - 9

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Logic for this lecture:

Same as in previous part…

Questions Discussed in this lecture:

Q68. $\int_{0}^{\frac{\pi}{2}}\sqrt{sin{\phi}}{cos}^5{\phi}d\phi=\frac{64}{231}$

Q70.  $\int_{0}^{1}\left[xe^x+sin{\frac{\pi\ x}{4}}\right]dx=1+\frac{4}{\pi}-\frac{2\sqrt2}{\pi}$

Q71.  $\int_{0}^{\frac{\pi}{2}}{{sin}^3{x}dx=\frac{2}{3}}$

Q72.  $\int{\frac{1}{\sqrt{{sin}^3{x}{cos}^5{x}}}dx=-\frac{2}{\sqrt{tan{x}}}+\frac{2}{3}(tan{x})^\frac{3}{2}+C}$

Q73.  $\int_{0}^{\frac{\pi}{4}}{2{tan}^3{x}dx}=1-log{2}$

PART - 10

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Logic for this lecture:

Second way: in case of all linear variables in question, you can substitute bigger linear expression of question and find the value for smaller linear expression from that. You can also use this method in case of exponential functions.
You can also use division if there is no root in function and it is an improper fraction.

Questions Discussed in this lecture:

Q74. $\int{\frac{x}{\sqrt{x+4}}dx=\frac{2}{3}\sqrt{x+4}(x-8)+C}$

Q75. $\int{\frac{2x-1}{2x+3}dx} = x-log{|}(2x+3)^2|+C$

Q76. $\int{\frac{8x+13}{\sqrt{4x+7}}dx} = \frac{1}{3}(4x+7)^\frac{3}{2}-\frac{1}{2}(4x+7)^\frac{1}{2}+C$

Q77. $\int{\frac{x}{\sqrt{x+2}}dx} = \frac{2}{3}(x+2)^\frac{3}{2}-4(x+1)^\frac{1}{2}+C$

Q78. $\int{\frac{x+1}{\sqrt{2x-1}}dx}=\frac{1}{6}(2x-1)^\frac{3}{2}+\frac{3}{2}(2x-1)^\frac{1}{2}+C$

PART - 11

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Logic for this lecture:

Third way: You can substitute any trigonometric or algebraic value.

Questions Discussed in this lecture:

Q79. $\int\frac{x^5}{x+1}dx=\frac{x^5}{5}-\frac{x^4}{4}+\frac{x^3}{3}-\frac{x^2}{2}+x-log{|}x+1|+C$

Q80. $\int{\frac{x^4+1}{x^2+1}dx=\frac{x^3}{3}-x+2{tan}^{-1}{x}+C}$

Q81. $\int{\frac{1}{x^\frac{1}{2}+x^\frac{1}{3}}dx=2\sqrt\ x-3x^\frac{1}{3}+6x^\frac{1}{6}-6log{(}1+x^\frac{1}{6})+C}$

Q82. $\int\sqrt{\frac{1-\sqrt\ x}{1+\sqrt\ x}}dx=-2\sqrt{1-x}+{cos}^{-1}{\sqrt\ x}+\sqrt{x-x^2}+C$

:: Updated 17th July, 2019 ::

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