Mathematics

# Evaluate the following integral$\int { \cfrac { \sin { \left( x+a \right) } }{ \sin { \left( x+b \right) } } } dx$

##### SOLUTION
$\displaystyle\int{\dfrac{\sin{\left(x+a\right)}}{\sin{\left(x+b\right)}}dx}$

$=\displaystyle\int{\dfrac{\sin{\left[\left(x+b\right)+\left(a-b\right)\right]}}{\sin{\left(x+b\right)}}dx}$

$=\displaystyle\int{\dfrac{\sin{\left(x+b\right)}\cos{\left(a-b\right)}+\cos{\left(x+b\right)}\sin{\left(a-b\right)}}{\sin{\left(x+b\right)}}dx}$

$=\cos{\left(a-b\right)}\displaystyle\int{dx}+{\sin{\left(a-b\right)}}\displaystyle\int{\dfrac{\cos{\left(x+b\right)}}{\sin{\left(x+b\right)}}dx}$

$=x\cos{\left(b-a\right)}-\sin{\left(b-a\right)}\log{\left|\sin{\left(x+b\right)}\right|}+C$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

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