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$\mathrm{\int_{0}^{20 \pi}(|\sin x|+|\cos x|)^{2} d x}$ is equal to
Option: 1
$\mathrm{10(\pi+4)}$

Option: 2
$\mathrm{10(\pi+2)}$

Option: 3
$\mathrm{20(\pi-2)}$

Option: 4
$\mathrm{20(\pi+2)}$

Answers (1)

best_answer

$\mathrm{|\sin x|+|\cos x|}$ is periodic with period $\mathrm{\frac{\pi}{2}}$

$\therefore$ Required expression

$\mathrm{=40 \int_{0}^{\pi / 2}(|\sin x|+|\cos x|)^{2} d x} \\$

$\mathrm{=40 \int_{0}^{\pi / 2}(\sin x+\cos x)^{2} d x}$

$\mathrm{=40 \int_{0}^{\pi / 2}(1+\sin 2 x) d x} \\$

$\mathrm{=40\left[x-\left.\frac{\cos 2 x}{2}\right|_{0} ^{\pi / 2}\right] }\\$

$=40\left[\frac{\pi}{2}-\frac{1}{2}(-1-1)\right] \\$

$=40\left[\frac{\pi}{2}+1\right] \\$

$=20(\pi+2)$

Hence correct option is 4

Posted by

Divya Prakash Singh

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