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Q. The value of the integral $\int\limits_{0}^{\pi / 4} \frac{\sin x+\cos x}{3+\sin 2 x} d x$ is equal to

WBJEEWBJEE 2012

Solution:

Let $I=\int\limits_{0}^{\pi / 4} \frac{\sin x+\cos x}{3+\sin 2 x} d x$
$=\int\limits_{0}^{\pi / 4} \frac{\sin x+\cos x}{3+2 \sin x \cos x} d x$
$=\int\limits_{0}^{\pi / 4}-\frac{\sin x+\cos x}{(\sin x-\cos x)^{2}-4} d x$ ...(i)
Put $\sin x-\cos x=t$
$\Rightarrow (\cos x+\sin x) d x=d t$
when $x=0 \Rightarrow t=-1$
and $x=\frac{\pi}{4} \Rightarrow t=0$
$\therefore $ Eq.(i) becomes,
$I=-\int\limits_{-1}^{0} \frac{d t}{t^{2}-4}$
$=-\frac{1}{4}\left[\log \left|\frac{t-2}{t+2}\right|\right]_{-1}^{0}$
$=-\frac{1}{4}(\log 1-\log 3)$
$=-\frac{1}{4}(0-\log 3)=\frac{1}{4} \log 3$