I= ^ 2 _0 (2+ )(1+ ) dx equals: - Vidyadip

MCQ

$\text{I}=\int\limits^\frac{\pi}{2}_0\frac{\cos\text{x}}{(2+\sin\text{x})(1+\sin\text{x})}\text{dx}$ equals:

A
$\log2-1$
B
$\log2$
C
$\log4-1$
✓
$\log\frac{4}{3}$

✓

Answer

Correct option: D.

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

Let, $\text{I}=\int\limits^\frac{\pi}{2}_0\frac{\cos\text{x}}{(2+\sin\text{x})(1+\sin\text{x})}\text{dx}$
Let $\sin\text{x}$ then $\cos\text{x}\text{ dx}=\text{dt}$
When $\text{x}=0,\text{t}=0,\text{x}=\frac{\pi}{2},\text{t}=1$
Therefore the integral becomes
$\text{I}=\int\limits^1_0\frac{\text{dt}}{(2+\text{t})(1+\text{t})}$
$=\int\limits^1_0\Big[\frac{-1}{2+\text{t}}+\frac{1}{1+\text{t}}\Big]\text{dt}$
$=\big[-\log(2+\text{t})+\log(1+\text{t})\big]^1_0$
$= \big[\log(1+\text{t})-\log(2+\text{t})\big]^1_0$
$=\log2-\log3-\log1+\log2$
$=\log\frac{4}{3}$

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