The value of $\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}}(\sin|x|+\cos|x|) dx$ is

4

The correct answer is Option (3) → 4

Given integral:

$I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left( \sin|x| + \cos|x| \right) dx$

Since $\sin|x|$ is even and $\cos|x|$ is even, the integrand is even.

$I = 2 \int_{0}^{\frac{\pi}{2}} \left( \sin x + \cos x \right) dx$

Integrating:

$I = 2 \left[ -\cos x + \sin x \right]_{0}^{\frac{\pi}{2}}$

$I = 2 \left[ (-\cos \frac{\pi}{2} + \sin \frac{\pi}{2}) - (-\cos 0 + \sin 0) \right]$

$I = 2 \left[ (0 + 1) - (-1 + 0) \right]$

$I = 2 \left[ 1 + 1 \right]$

$I = 4$