Lecture 1 Part 5 Integrals Class 12 Maths

Part - 5

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

Integrals Lecture 1 Part 5
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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