MCQ
If $I=\int \frac{d x}{\sin (x-a) \sin (x-b)}$, then $I$ is given by
- A
$\frac{1}{\sin (b-a)} \log |\sin (x-a) \sin (x-b)|+c,$ where $c$ is a constant of integration.
- B
- C
- D
$\frac{1}{\sin (b-a)} \log |\sin (x-a) \sin (x-b)|+c,$ where $c$ is a constant of integration.
$\begin{aligned} & \text {(d) : Let } I=\int \frac{d x}{\sin (x-a) \sin (x-b)} \\ & =\frac{1}{\sin (b-a)} \int \frac{\sin (b-a)}{\sin (x-a) \sin (x-b)} d x\end{aligned}$
$\begin{aligned} & =\frac{1}{\sin (b-a)} \int \frac{\sin ((x-a)-(x-b))}{\sin (x-a) \sin (x-b)} d x \\ & =\frac{1}{\sin (b-a)} \int \frac{\sin (x-a) \cos (x-b)-\cos (x-a) \sin (x-b)}{\sin (x-a) \sin (x-b)} d x \\ & =\frac{1}{\sin (b-a)} \int(\cot (x-b)-\cot (x-a)) d x \\ & =\frac{1}{\sin (b-a)}[\log |\sin (x-b)|-\log |\sin (x-a)|]+c \\ & =\frac{1}{\sin (b-a)} \log \left|\frac{\sin (x-b)}{\sin (x-a)}\right|+c\end{aligned}$
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.