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

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

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

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

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

JEE Main · 2008 · 27 Apr 2008 · Offline · Indefinite Integrals

The value of

2 \int \frac{s i n x d x}{s i n ( x - \frac{π}{4} )}

Answer: (C)

Write the numerator's angle in terms of the denominator's:

sin x = sin ((x - \frac{π}{4}) + \frac{π}{4}) = sin (x - \frac{π}{4}) cos \frac{π}{4} + cos (x - \frac{π}{4}) sin \frac{π}{4}

. Dividing by

sin (x - \frac{π}{4})

and multiplying by

2

turns the integrand into

1 + cot (x - \frac{π}{4})

, and integrating gives

x + ln sin (x - \frac{π}{4}) + c

JEE Main · 2008 · 27 Apr 2008 · Offline · Single correct option