The value of $\int\limits_{-\frac{π}{2}}^{\frac{π}{2}}(\sin|x|+\cos|x|)dx$, is equal to:

4

The correct answer is Option (4) → 4

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

$\sin|x|$ and $\cos|x|$ are both even functions.

So, $\int_{-a}^{a} f(x)\,dx = 2 \int_{0}^{a} f(x)\,dx$ for even functions.

Hence, the integral becomes:

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

$= 2 \left[ -\cos x + \sin x \right]_0^{\frac{\pi}{2}}$

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

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

$= 2 (1 + 1) = 4$