Question

$\int \frac{\sin x}{\sin (x-\alpha)} d x$ is

(a) $x \sin \alpha+\cos \alpha \log \sin (x+\alpha)+C$

(b) $x \sin \alpha+\cos \alpha \log \sin (x-\alpha)+C$

(c) $x \cos \alpha+\sin \alpha \log \cos (x+\alpha)+C$

(d) None of the above

Answer 1

SOLUTION — Put $x-\alpha=t \Rightarrow d x=d t$

$\begin{aligned}\therefore \quad I & =\int \frac{\sin (\alpha+t)}{\sin t} d t=\sin \alpha \int \cot t d t+\cos \alpha \int d t \\& =\sin \alpha \log \sin t+\cos \alpha \cdot t+C_1 \\& =\sin \alpha \log \sin (x-\alpha)+x \cos \alpha+C \\& \left.\quad \text { (let } C=-\alpha \cos \alpha+C_1\right)\end{aligned}$

So, The correct option of this question will be (D).