L1 – Direct Integration

Formulas 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 \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}

PART - 1

Integrals Lecture 1 Part 1
Play Video

Logics for this lecture:
Question must be in addition/subtraction form.

Try to convert division/multiplication between functions to addition/subtraction by simplification.

If there are functions other than algebraic functions like trigonometry and logarithmic function, then you can use their respective identities to simplify them. Some of the identities are:
1+\cos x=2\cos ^{2} \frac{x}{2}

1-\cos x=2\sin ^{2} \frac{x}{2}

1\pm \sin x=\left(\cos \frac{x}{2} \pm \sin \frac{x}{2} \right)^{2}

\frac{1+\tan x}{1-\tan x} =\tan \left(\frac{\pi }{4} +x\right)

\frac{1-\tan x}{1+\tan x} =\tan \left(\frac{\pi }{4} -x\right)

\log e^{x} =x

e^{\log x} =x

Questions Discussed in this lecture:
1. \int x^{2} \left(1-\frac{1}{x^{2} } \right)\, dx =\frac{x^{3} }{3} -x+C

2. \int _{}^{}\left(\sqrt{x} -\frac{1}{\sqrt{x} } \right) ^{2} dx=\frac{x^{2} }{2} +\log |x|-2x+C

3. \int \left(8^{x} +x^{8} +\frac{8}{x} +\frac{x}{8} \right)\, dx =\frac{8^{x} }{\log 8} +\frac{x^{9} }{9} +8\log |x|+\frac{x^{2} }{16} +C

4. \int (e^{a\log x} +e^{x\log a} )\, dx =\frac{x^{a+1} }{a+1} +\frac{a^{x} }{\log a} +C

5. \int \left(\frac{\cos 2x+2\sin ^{2} x}{\cos ^{2} x} \right) \, dx=\tan x+C

6. \int (x^{c} +c^{x} )\, dx =\frac{x^{c+1} }{c+1} +\frac{c^{x} }{\log c} +C

7. \int _{}^{}\frac{x^{2} +3x+4}{\sqrt{x} } \, dx=\frac{2}{5} x^{\frac{5}{2} } +2x^{\frac{3}{2} } +8\sqrt{x} +C

8. \int \left(\frac{2a}{\sqrt{x} } -\frac{b}{x^{2} } +3c\, \, \sqrt[{3}]{x^{2} } \right) \, dx=4a\sqrt{x} +\frac{b}{x} +\frac{9cx^{\frac{5}{3} } }{5} +C

9. \int _{}^{}\frac{x^{3} -x^{2} +x-1}{x-1} \, \, dx =\frac{x^{3} }{3} +x+C

10. \int _{}^{}\frac{\sec ^{2} x}{{\rm cosec}^{2} \, x} \, dx=\tan x-x+C

11. \int \sqrt{1+\sin 2x} \, dx=-\cos x+\sin x+C

PART - 2

Integrals Lecture 1 Part 2
Play Video

Logic for this lecture:
You can use all integration identities directly if there is linear function in place of x, the only extra thing you need to do in this case is to write derivative of that linear function in denominator of final answer.
Why?? Reason is in the video lecture!

Questions Discussed in this lecture:
12. \int _{}^{}\sin 2x.dx =-\frac{1}{2} \cos 2x+C

13. \int \frac{1}{\sqrt{x} +\sqrt{x-1} } \, dx =\frac{2}{3} x^{\frac{3}{2} } -\frac{2}{3} (x-1)^{\frac{3}{2} } +C

14. \int \left(x-\frac{1}{2} \right) ^{3} dx=\frac{1}{4} \left(x-\frac{1}{2} \right)^{4} +C

15. \int \frac{e^{x} }{a^{x} }  \, dx=\frac{\left(\frac{e}{a} \right)^{x}}{log(e/a)} +C

16. \int \cos ^{2} \alpha  \, dx=x\cos ^{2} \alpha +C

17. \int \frac{1}{x\cos \alpha +1}  \, dx=\frac{\log |x\cos \alpha +1|}{\cos \alpha } +C

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

19. \int \frac{\sin x+\cos x}{\sqrt{1+\sin 2x} } \, dx =x+C

20. \int (\sin ^{-1}  \sqrt{x} +\cos ^{-1} \sqrt{x} \, )\, dx=\frac{\pi x}{2} +C

21. \int \frac{1}{1-\sin ^{2} x}  \, dx=\tan x+C

22. \int _{}^{}\frac{2-3\sin x}{\cos ^{2} x}  \, dx=2\tan x-3\sec x+C

23. \int e^{-\log e^{x} } dx =-e^{-x} +C

24. \int \sqrt{1+\cos 2x}  \, dx=\sqrt{2} \sin x+C

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

26. \int \tan ^{-1} (\sec x+\tan x) \, dx=\frac{\pi }{4} x+\frac{x^{2} }{4} +C

27. \int \tan ^{-1} \sqrt{\frac{1-\sin x}{1+\sin x} } dx =\frac{\pi }{4} x-\frac{x^{2} }{4} +C

28. \int \left(\sqrt{ax} -\frac{1}{\sqrt{ax} } \right) ^{2} dx=\frac{ax^{2} }{2} +\frac{\log |x|}{a} -2x+C

29. \int \frac{\sin 4x}{\sin 2x} \, dx =  \sin 2x+C

30. \int _{}^{}e^{2x+3} dx = \frac{1}{2} e^{2x+3} +C

31. \int _{}^{}\sec ^{2} (7-4x)dx = -\frac{1}{4} \tan (7-4x)+C

PART - 3

Integrals Lecture 1 Part 3
Play Video

Logic for this lecture:
Same logic we have used in Part 2 of this lecture!

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

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

34. \int _{}^{}\sin ^{2} (2x+5)\, dx = \frac{x}{2} -\frac{1}{8} \sin (4x+10)+C

35. \int _{}^{}\sin ^{3} (2x+1)\, dx   =  -\frac{1}{2} \cos (2x+1)+\frac{1}{6} \cos ^{3} (2x+1)+C

36. \int _{}^{}\sin ^{4}  x\, dx  =  \frac{3x}{8} -\frac{1}{4} \sin 2x+\frac{1}{32} \sin 4x+C

37. \int \sin x.\sin 2x\, dx    =  -\frac{1}{2} \left(\frac{\sin 3x}{3} -\sin x\right)+C

38. \int _{}^{}\sin 3x\cos 4x \, dx  =  -\frac{1}{14} \cos 7x+\frac{1}{2} \cos x+C

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

40. \int \sin ^{2} x\, \, \cos ^{4} x \, dx  =  \frac{1}{32} \left[2x+\frac{1}{2} \sin 2x-\frac{1}{2} \sin 4x-\frac{1}{6} \sin 6x\right]+C

41. \int _{}^{}\cos 2x\cos 4x\cos 6x \, dx = \frac{1}{4} \left[\frac{1}{12} \sin 12x+x+\frac{1}{8} \sin 8x+\frac{1}{4} \sin 4x\right]+C

43. \int _{}^{}\sin x\sin 2x\sin 3x\, dx  = \frac{1}{4} \left[\frac{1}{6} \cos 6x-\frac{1}{4} \cos 4x-\frac{1}{2} \cos 2x\right]+C

44. \int \tan ^{-1} \sqrt{\frac{1-\sin x}{1+\sin x} } \, dx = \frac{\pi }{4} x-\frac{x^{2} }{4} +C

45. \int _{}^{}\frac{e^{5\log x} -e^{4\log x} }{e^{3\log x} -e^{2\log x} } \, dx    =  \frac{x^{3} }{3} +C

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

47. \int \tan ^{-1} (\cot x)\, dx =  \frac{\pi }{2} x-\frac{x^{2} }{2} +C

48. \int _{0}^{\pi }\, ( \sin ^{2} \frac{x}{2} -\cos ^{2} \frac{x}{2} )\, dx  =  0

49. \int _{0}^{1}\frac{dx}{\sqrt{1+x} -\sqrt{x} } = \frac{4\sqrt{2} }{3}

PART - 4

Integrals Lecture 1 Part 4
Play Video

Logic for this lecture:
You can rationalise non-root functions also

Questions Discussed in this lecture:
49. \int \frac{1}{1+\cos x}  \, dx=\tan \frac{x}{2} +C

50. \int \frac{1}{1+\sin x} \, dx =\tan x-\sec x+C

51. \int \frac{\sin x}{1+\sin x} \, dx =\sec x-\tan x+x+C

52. \int _{}^{}\frac{\cos x}{1+\cos x}  \, dx  = x-\tan \frac{x}{2} +C

PART - 5

Integrals Lecture 1 Part 5
Play Video

Logic for this lecture:
In case trigonometric functions are given as a fraction you can use following identities in numerator to simplify it:
\sin (A\pm B)=\sin A\cos B\pm \cos A\sin B
\cos (A\pm B)=\cos A\cos B\mp \sin A\sin B

Questions Discussed in this lecture:
53. \int \frac{\sin (x-\alpha )}{\sin (x+\alpha )}  \, dx=x\cos 2\alpha -\sin 2\alpha \log |\sin (x+\alpha )|+C

54. \int \frac{\cos (x+a)}{\cos (x-a)} \, dx =x\cos 2a-\sin 2a\, \log |\sec (x-a)|+C

55. \int _{}^{}\frac{\sin x}{\sin (x+a)} dx  = x\cos a-\sin a.\log |\sin (x+a)|+C

56. \int \frac{1}{\sin (x-a)\sin (x-b)} dx   = \frac{1}{\sin (a-b)} \log \left|\frac{\sin (x-a)}{\sin (x-b)} \right|+C

57. \int _{}^{}\frac{\sin x}{\sin (x-a)} \, dx  =  \sin a\log |\sin (x-a)|+x\cos a+C

58. \int _{}^{}\frac{1}{\cos (x+a)\cos (x+b)}  \, dx   =  \frac{1}{\sin (a-b)} \log \left|\frac{\cos (x+b)}{\cos (x+a)} \right|+C

59. \int \frac{\sin (x+a)}{\sin (x+b)} \, dx   =  x\cos (a-b)+\sin (a-b)\log |\sin (x+b)|+C

60. \int \frac{\cos (x+a)}{\sin (x+b)} \, dx  = \cos (a-b)\, \log |\sin (x+b)|-\sin (a-b).x+C

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