# The value of is $\dpi{100} \sqrt{2}\int \frac{\sin x\, dx}{\sin \left ( x-\frac{\pi }{4} \right )}$ Option 1) $x-\log \left | \cos \left ( x-\frac{\pi }{4} \right ) \right |+c$ Option 2) $x+\log \left | \cos \left ( x-\frac{\pi }{4} \right ) \right |+c$ Option 3) $x-\log \left | \sin \left ( x-\frac{\pi }{4} \right ) \right |+c$ Option 4) $x+\log \left | \sin \left ( x-\frac{\pi }{4} \right ) \right |+c$

P perimeter

As learnt in the concept

Integrals for Trigonometric functions -

$\frac{\mathrm{d} }{\mathrm{d} x}\left ( -cos x \right ) =sinx$

$\therefore \int sinxdx=-cosx+c$

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$\sqrt{2}\int \frac{sinxdx}{sin(x-\frac{\pi}{4})}$

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

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

$=\sqrt{2}\int \frac{1 }{\sqrt{2}}dx +\sqrt{2}\int \frac{1}{\sqrt{2}}cot\left (x-\frac{\pi }{4} \right )dx$

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

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

Option 1)

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

Option 2)

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

Option 3)

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

Option 4)

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

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