Mathematics

If $$\int { \cfrac { \sin { x }  }{ \sin { \left( x-\alpha  \right)  }  }  } dx=Ax+B\log { \sin { \left( x-\alpha  \right)  }  } +c$$, then the value of $$(A,B)$$ is-


ANSWER

$$\left( \cos { \alpha } ,\sin { \alpha } \right) $$


SOLUTION
$$\displaystyle \int \frac{sin(x-\alpha +\alpha )}{sin(x-\alpha )}dx$$
$$ \displaystyle \int \frac{sin(x-\alpha )cos\alpha +cos(x-\alpha )sin\alpha }{sin(x-\alpha )}$$
$$ \displaystyle \int (cos\alpha +sin\alpha \,cot(x-\alpha ))dx$$
$$  xcos\alpha +sin\alpha ln sin(x-\alpha )+c$$
$$ A = cos\alpha $$
$$ B = sin\alpha $$

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Single Correct Medium Published on 17th 09, 2020
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