∫1sin⁡(x−a)sin⁡(x−b)dx=

$\int \frac{1}{\mathrm{sin}\left(x-a\right)\mathrm{sin}\left(x-b\right)}dx=$

1. A

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

2. B

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

3. C

$\mathrm{log}\mathrm{sin}\left(x-a\right)\mathrm{sin}\left(x-b\right)+C$

4. D

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

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Solution:

We have,

$\begin{array}{l}\int \frac{1}{\mathrm{sin}\left(x-a\right)\mathrm{sin}\left(x-b\right)}dx\\ =\frac{1}{\mathrm{sin}\left(b-a\right)}\int \frac{\mathrm{sin}\left\{\left(x-a\right)-\left(x-b\right)\right\}}{\mathrm{sin}\left(x-a\right)\mathrm{sin}\left(x-b\right)}dx\\ =\frac{1}{\mathrm{sin}\left(b-a\right)}\int \left\{\mathrm{cot}\left(x-b\right)-\mathrm{cot}\left(x-a\right)\right\}dx\\ =\frac{1}{\mathrm{sin}\left(b-a\right)}\left\{\mathrm{log}\mathrm{sin}\left(x-b\right)-\mathrm{log}\mathrm{sin}\left(x-a\right)\right\}+C\\ =\frac{1}{\mathrm{sin}\left(b-a\right)}\mathrm{log}\left|\frac{\mathrm{sin}\left(x-b\right)}{\mathrm{sin}\left(x-a\right)}\right|+C\\ =\frac{1}{\mathrm{sin}\left(a-b\right)}\mathrm{log}\left|\frac{\mathrm{sin}\left(x-a\right)}{\mathrm{sin}\left(x-b\right)}\right|+C\end{array}$

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