Q39 / 90 · JEE Main 2022 (26 Jul Shift 2) · Mathematics

$\int_{0}^{20 π} {(|\sin x| + |\cos x|)}^{2} d x$ is equal to:

A
$10 (π + 4)$
B
$10 (π + 2)$
C
$20 (π - 2)$
Correct answer
D
$20 (π + 2)$

Solution

Let $I = \int_{0}^{20 π} {(|\sin x| + |\cos x|)}^{2} d x$
We know that the period of $|\sin x| + |\cos x|$ is $\frac{π}{2}$
i.e. $I = 40 \int_{0}^{\frac{π}{2}} {(\sin x + \cos x)}^{2} d x$
$I = 40 \int_{0}^{\frac{π}{2}} (1 + \sin 2 x) d x$
$= 40 {(x - \frac{\cos 2 x}{2})}_{0}^{\frac{π}{2}} = 40 (\frac{π}{2} - \frac{cosπ}{2} + \frac{\cos 0}{2})$
$I = 20 [π + 2]$

Hence, the correct answer is option: 'D'.

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Solution

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