_ ^ x (1 + x)(2 + x) ;dx = - Vidyadip

MCQ

$\int_{}^{} {\frac{{\cos x}}{{(1 + \sin x)(2 + \sin x)}}\;dx = } $

A
$\log [(1 + \sin x)(2 + \sin x)] + c$
B
$\log \frac{{2 + \sin x}}{{1 + \sin x}} + c$
✓
$\log \frac{{1 + \sin x}}{{2 + \sin x}} + c$
D
None of these

✓

Answer

Correct option: C.

$\log \frac{{1 + \sin x}}{{2 + \sin x}} + c$

c
(c) Put $\sin x = t \Rightarrow \cos x\,dx = dt,$ then
$\int_{}^{} {\frac{{\cos x}}{{(1 + \sin x)(2 + \sin x)}}} \,dx = \int_{}^{} {\frac{{dt}}{{(t + 1)(t + 2)}}} $$ = \int_{}^{} {\frac{1}{{t + 1}}dt - \int_{}^{} {\frac{1}{{t + 2}}dt} } = \log \left( {\frac{{t + 1}}{{t + 2}}} \right) + c = \log \left( {\frac{{\sin x + 1}}{{\sin x + 2}}} \right) + c$.

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