First slide

Methods of integration

Question

$\int \frac{1}{\sin (x - a) \sin (x - b)} d x =$

Moderate

A

$\frac{1}{\sin (a - b)} \log |\frac{\sin (x - a)}{\sin (x - b)}| + C$
B

$- \frac{1}{\sin (a - b)} \log |\frac{\sin (x - a)}{\sin (x - b)}| + C$
C

$\log \sin (x - a) \sin (x - b) + C$
D

$\log |\frac{\sin (x - a)}{\sin (x - b)}| + C$

Solution

We have,
$\begin{array}{l} \int \frac{1}{\sin (x - a) \sin (x - b)} d x \\ = \frac{1}{\sin (b - a)} \int \frac{\sin {(x - a) - (x - b)}}{\sin (x - a) \sin (x - b)} d x \\ = \frac{1}{\sin (b - a)} \int {\cot (x - b) - \cot (x - a)} d x \\ = \frac{1}{\sin (b - a)} {\log \sin (x - b) - \log \sin (x - a)} + C \\ = \frac{1}{\sin (b - a)} \log |\frac{\sin (x - b)}{\sin (x - a)}| + C \\ = \frac{1}{\sin (a - b)} \log |\frac{\sin (x - a)}{\sin (x - b)}| + C \end{array}$

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