Question

integrate
some one pls help its urgent ​

Answer 1

Answer:

$∫ \frac{ \sqrt{2} \sin(x) }{ \sin(x - \frac{\pi}{4} ) } d(x)$

$= \sqrt{2} ∫ \frac{ \sin(x) }{ \sin(x - \frac{\pi}{4} ) } d(x)$

$= \sqrt{2} ∫ \frac{ \sin(x - \frac{\pi}{4} + \frac{\pi}{4} ) }{ \sin(x - \frac{\pi}{4} ) }$

$= \sqrt{2} ∫[\cos( \frac{\pi}{4} ) + \cot(x - \frac{\pi}{4} ) \sin( \frac{\pi}{4} ) ]d(x)$

$= \sqrt{2} . \frac{1}{ \sqrt{2} } x + \sqrt{2} . \frac{1}{ \sqrt{2} } ln | \sin(x - \frac{\pi}{4} ) | + c$

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