0 / 2 cos x ( 1 + sin x ) ( 2 + sin x ) ? d x =

Question

$\int_{0}^{π / 2} \frac{\cos x}{(1 + \sin x) (2 + \sin x)} d x =$

Smart Achievers · Accepted Answer

(1)

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

Explanation:

Put $\sin x = t$ $\Rightarrow \cos x d x = d t,$ at $x = 0, t = 0$

at $x = \frac{π}{2}, t = 1$

So that the reduced integral is

$\int_{0}^{1} \frac{d t}{(1 + t) (2 + t)} = \int_{0}^{1} (\frac{1}{1 + t} - \frac{1}{2 + t}) d t$

$= {[\log (1 + t) - \log (2 + t)]}_{0}^{1}$

$= [\log 2 - \log 3] - [\log 1 - \log 2]$

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

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