2 sinx ;dx sin ( x - 4 ) = - Vidyadip

MCQ

$\sqrt 2 \smallint \frac{{sinx\;dx}}{{{\rm{sin}}\left( {x - \frac{\pi }{4}} \right)}} = $

A
$x + \log \left| {\cos \left( {x - \frac{\pi }{4}} \right)} \right| + c\;$
B
$\;x - \log \left| {\sin \left( {x - \frac{\pi }{4}} \right)} \right| + c$
✓
$\;x + \log \left| {\sin \left( {x - \frac{\pi }{4}} \right)} \right| + c$
D
$\;x - \log \left| {\cos \left( {x - \frac{\pi }{4}} \right)} \right| + c$

✓

Answer

Correct option: C.

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

c
$\sqrt{2} \int \frac{\sin x d x}{\sin \left(x-\frac{\pi}{4}\right)}$$=\sqrt{2} \int \frac{\sin \left(x-\frac{\pi}{4}+\frac{\pi}{4}\right) d x}{\sin \left(x-\frac{\pi}{4}\right)}$

$=\sqrt{2} \int\left(\cos \frac{\pi}{4}+\cot \left(x-\frac{\pi}{4}\right) \sin \frac{\pi}{4}\right) d x$

$=\int d x+\int \cot \left(x-\frac{\pi}{4}\right) d x$

$=x+\ln \left|\sin \left(x-\frac{\pi}{4}\right)\right|+c$

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