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The value of $\int_{0}^{π / 2} \frac{1 + 2 \cos x}{(2 + \cos x)^{2}} d x$ is

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Correct option is C

∫0π/2 1+2cos⁡x(2+cos⁡x)2dx=∫0π/2 2(cos⁡x+2)−3(2+cos⁡x)2dx=4∫01 dt3+t2−6∫01 1+t23+t22dt (t=tan⁡x/2)=−2∫01 dt3+t2+12∫01 dt3+t22=−2∫01 dt3+t2+1212.3tt2+301+12.3∫01 dt3+t2=12.

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The value of ∫0π/2 1+2cos⁡x(2+cos⁡x)2dx is

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The value of $\int_{0}^{π / 2} \frac{1 + 2 \cos x}{(2 + \cos x)^{2}} d x$ is