Mathematics

Evaluate the following integral
$$\int { \cfrac { \sin { \left( x-\alpha \right)  }  }{ \sin { \left( x+\alpha \right)  }  }  } dx$$


SOLUTION
$$\displaystyle\int{\dfrac{\sin{\left(x-\alpha\right)}}{\sin{\left(x+\alpha\right)}}dx}$$

$$=\displaystyle\int{\dfrac{\sin{\left(x+\alpha-2\alpha\right)}}{\sin{\left(x+\alpha\right)}}dx}$$

$$=\displaystyle\int{\dfrac{\sin{\left(x+\alpha\right)}\cos{2\alpha}-\cos{\left(x+\alpha\right)}\sin{2x}}{\sin{\left(x+\alpha\right)}}dx}$$

$$=\displaystyle\int{\dfrac{\sin{\left(x+\alpha\right)}\cos{2\alpha}}{\sin{\left(x+\alpha\right)}}dx}-\displaystyle\int{\dfrac{\cos{\left(x+\alpha\right)}\sin{2x}}{\sin{\left(x+\alpha\right)}}dx}$$

$$=\cos{2\alpha}\displaystyle\int{dx}-\sin{2\alpha}\displaystyle\int{\dfrac{\cos{\left(x+\alpha\right)}}{\sin{\left(x+\alpha\right)}}dx}$$

$$=x\cos{2\alpha}-\sin{2\alpha}\log{\left|\sin{\left(x+\alpha\right)}\right|}+c$$

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Subjective Medium Published on 17th 09, 2020
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