Mathematics

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


SOLUTION
$$\int {\dfrac{{\sin \left( {x + a} \right)}}{{\sin \left( {x + b} \right)}}} dx$$
$$putting\,x + b = t$$
$$dx = dt$$
$$ = \int {\dfrac{{\sin \left( {t - b + a} \right)}}{{\sin t}}} dt$$
$$ = \int {\dfrac{{\sin \left( {t + \left( {a - b} \right)} \right)}}{{\sin t}}} dt$$
$$ = \int {\left[ {\dfrac{{\sin t\cos \left( {a - b} \right)}}{{\sin t}} + \dfrac{{\cos t\sin \left( {a - b} \right)}}{{\sin t}}} \right]} dt$$
$$ = \int {\cos \left( {a - b} \right)dt + \sin \left( {a - b} \right)\int {\cot tdt} } $$
$$ = \cos \left( {a - b} \right)dt + \sin \left( {a - b} \right)\log \sin \left| t \right| + c$$
$$ = \cos \left( {a - b} \right)dt + \sin \left( {a - b} \right)\log \sin \left| {x + b} \right| + c$$
$$ = \left( {x + b} \right)\cos \left( {a - b} \right) + \sin \left( {a - b} \right)\log \sin \left| {x + b} \right| + c$$
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Subjective Medium Published on 17th 09, 2020
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