Question

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

• A. $(\sin \alpha ,\cos \alpha )$
• B. $(\cos \alpha ,\sin \alpha )$
• C. $(-\sin \alpha ,\cos \alpha )$
• D. $(-\cos \alpha ,\sin \alpha )$

Answer 1

The correct option is B $(\cos \alpha ,\sin \alpha )$

$\int \frac{sin(x-\alpha +\alpha )}{sin(x-\alpha )}dx$

$\int \frac{sin(x-\alpha )cos\alpha +cos(x-\alpha )sin\alpha }{sin(x-\alpha )}$

$\int (cos\alpha +sin\alpha cot(x-\alpha \left)\right)dx$

$xcos\alpha +sin\alpha lnsin(x-\alpha )+c$

$A=cos\alpha$

$B=sin\alpha$