_0^ 1 1+ x dx equals - Vidyadip

Question

$\int_0^\pi \frac{1}{1+\sin x} dx$ equals

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Answer

$ \text { Explanation: } \int_0^\pi \frac{1}{1+\sin x} d x$
$=\int_0^\pi \frac{1}{1+\sin x} \times \frac{1-\sin x}{1-\sin x} d x$
$=\int_0^\pi \frac{1-\sin x}{1-\sin ^2 x} d x$
$=\int_0^\pi \frac{1-\sin x}{\cos ^2 x} d x$
$=\int_0^\pi\left(\sec ^2 x-\sec x \tan x\right) d x$
$=[\tan x-\sec x]_0^\pi$
$=0+1-0+1$
$=2 $

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