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
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