The value of  2∫sin⁡xsin⁡x−π4dx, is

# The value of  $\sqrt{2}\int \frac{\mathrm{sin}x}{\mathrm{sin}\left(x-\frac{\pi }{4}\right)}dx,$ is

1. A

$x+\mathrm{log}\left|\mathrm{sin}\left(x-\frac{\pi }{4}\right)\right|+C$

2. B

$x-\mathrm{log}\left|\mathrm{cos}\left(x-\frac{\pi }{4}\right)\right|+C$

3. C

$x+\mathrm{log}\left|\mathrm{cos}\left(x-\frac{\pi }{4}\right)\right|+C$

4. D

$x-\mathrm{log}\left|\mathrm{sin}\left(x-\frac{\pi }{4}\right)\right|+C$

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### Solution:

We have,

$\begin{array}{l}\sqrt{2}\int \frac{\mathrm{sin}x}{\mathrm{sin}\left(x-\frac{\pi }{4}\right)}dx\\ =\sqrt{2}\int \frac{\mathrm{sin}\left\{\left(x-\frac{\pi }{4}\right)+\frac{\pi }{4}\right\}}{\mathrm{sin}\left(x-\frac{\pi }{4}\right)}dx\\ =\sqrt{2}\int \frac{\mathrm{sin}\left(x-\frac{\pi }{4}\right)\mathrm{cos}\frac{\pi }{4}+\mathrm{cos}\left(x-\frac{\pi }{4}\right)\mathrm{sin}\frac{\pi }{4}}{\mathrm{sin}\left(x-\frac{\pi }{4}\right)}dx\\ =\int 1\cdot dx+\int \mathrm{cot}\left(x-\frac{\pi }{4}\right)dx=x+\mathrm{log}\left|\mathrm{sin}\left(x-\frac{\pi }{4}\right)\right|+C\end{array}$

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