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

MCQ

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

✓
$\log \frac{4}{3}$
B
$\log \frac{1}{3}$
C
$\log \frac{3}{4}$
D
None of these

✓

Answer

Correct option: A.

$\log \frac{4}{3}$

a
(a) Put $\sin x = t \Rightarrow \cos x\,dx = dt,$

so that reduced integral is

$\int_0^1 {\left( {\frac{1}{{1 + t}} - \frac{1}{{2 + t}}} \right)\,\,dt = [\log (1 + t) - \log (2 + t)]_0^1} $

$ = \log \frac{2}{3} - \log \frac{1}{2} = \log \frac{4}{3}$.

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