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

MCQ

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

✓
$\log \left(\frac{4}{3}\right)$
B
$\log \left(\frac{1}{3}\right)$
C
$\log \left(\frac{3}{4}\right)$
D
$\log \left(\frac{2}{3}\right)$

✓

Answer

Correct option: A.

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

(A)
Put $\sin x= t \Rightarrow \cos x d x= dt$
When $x=0, t =0$ and when $x=\frac{\pi}{2}, t =1$
$\therefore \quad \int_0^{\pi / 2} \frac{\cos x}{(1+\sin x)(2+\sin x)} d x$
$\begin{array}{l}=\int_0^1 \frac{ dt }{(1+ t )(2+ t )} \\ =\int_0^1\left(\frac{1}{1+ t }-\frac{1}{2+ t }\right) dt \\ =[\log (1+ t )-\log (2+ t )]_0^1 \\ =\log \left(\frac{2}{3}\right)-\log \left(\frac{1}{2}\right) \\ =\log \left(\frac{4}{3}\right)\end{array}$

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