MCQ
$\int \frac{1}{\sin (x-a) \sin x} d x=$
- A
- B
- C
- D$-\operatorname{cosec} a(\log (\sin (x-a) \cdot \sin x))+c$, where $c$ is a constant of integration
$\begin{aligned} & \text {(b) : Let } I=\int \frac{d x}{\sin (x-a) \sin x} \\ & =\int \frac{\sin a}{\sin a \cdot \sin x \sin (x-a)} d x \\ & =\frac{-1}{\sin a} \int \frac{\sin (x-a-x)}{\sin x \sin (x-a)} d x \\ & =\frac{-1}{\sin a} \int \frac{\sin (x-a) \cos x-\cos (x-a) \sin x}{\sin x \sin (x-a)} d x \\ & =\frac{-1}{\sin a} \int[\cot x-\cot (x-a)] d x\end{aligned}$
$\begin{aligned} & =\frac{-1}{\sin a}[\ln (\sin x)-\ln \sin (x-a)]+C \\ & =\operatorname{cosec} a\left[\ln \sin (x-a)-\ln \left(\frac{1}{\operatorname{cosec} x}\right)\right]+C \\ & =\operatorname{cosec} a[\ln (\sin (x-a) \cdot \operatorname{cosec} x]+C\end{aligned}$
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