_0^ / 2 x (2+ x)(1+ x) d x equals - Vidyadip

MCQ

$\int_0^{\pi / 2} \frac{\cos x}{(2+\sin x)(1+\sin x)} d x$ equals

A
$\log \left(\frac{3}{4}\right)$
B
$\log \left(\frac{3}{2}\right)$
✓
$\log \left(\frac{4}{3}\right)$
D
$\log \left(\frac{2}{3}\right)$

✓

Answer

Correct option: C.

$\log \left(\frac{4}{3}\right)$

Let $I=\int_0^{\frac{\pi}{2}} \frac{\cos x}{(2+\sin x)(1+\sin x)} d x$
Let $\sin x = t$ then $\cos x dx = dt$
When $x =0, t =0 x =\frac{\pi}{2}, t =1$
Therefore the integral becomes
$I=\int_0^1 \frac{d t}{(2+t)(1+t)}$
$=\int_0^1\left[\frac{-1}{2+t}+\frac{1}{1+t}\right] d t$
$=[-\log (2+t)+\log (1+t)]_0^1$
$=[\log (1+t)-\log (2+t)]_0^1$
$=\log 2-\log 3-\log 1+\log 2$
$=\log \frac{4}{3}$

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