# 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:
$\sin (A\pm B)=\sin A\cos B\pm \cos A\sin B$
$\cos (A\pm B)=\cos A\cos B\mp \sin A\sin B$

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