The correct option is B 2 π/2 ∫_0^πsin^2x(1 #10;+cos x)^2dx=∫1 cos^2x(1 +cos x)^2dx #10; ∫sin^2x(1 +cos x)^2dx=∫1 cos^2x(1 #10;+cos x)^2 ...

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Property 7

∫0π/2Sin2x/1+...

Question

$\int_{0}^{π_{/} 2} \frac{S i n^{2} x}{(1 + C o s x)^{2}} d_{X =}$

π / 2

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2 - π / 2

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π / 2 - 2

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2 + π / 2

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Solution

The correct option is B $2 - π / 2$
$\int_{0}^{π} \frac{s i n^{2} x}{(1 + c o s x)^{2}} d x = \int \frac{1 - c o s^{2} x}{(1 + c o s x)^{2}} d x$
$\int \frac{s i n^{2} x}{(1 + c o s x)^{2}} d x = \int \frac{1 - c o s^{2} x}{(1 + c o s x)^{2}} d x$
$= 2 \int t a n^{2} \frac{x}{2} \frac{d x}{2}$
$2 ⎡ ⎢ ⎣ \int_{0}^{\frac{π}{2}} (s e c^{2} \frac{x}{2} - 1) \frac{d x}{2} ⎤ ⎥ ⎦ = 2 ⎡ ⎢ ⎣ t a n \frac{x}{2} \int_{0}^{\frac{π}{2}} - \frac{x}{2} \int_{0}^{\frac{π}{2}} ⎤ ⎥ ⎦$
$= 2 [(1 - 0) - (\frac{π}{3} - 0)]$
$= 2 - \frac{π}{2}$

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