The value of $\int\limits_{-\pi/2}^{\pi/2}(x^5+x^3\cos x)dx$ is

0

The correct answer is Option (1) → 0

Integral: $\displaystyle \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (x^5 + x^3\cos x)\,dx$

$x^5$ is an odd function → integral over symmetric limits = $0$.

$x^3$ is odd and $\cos x$ is even → product $x^3\cos x$ is odd → integral over symmetric limits = $0$.

Total integral = $0 + 0 = 0$

The value of the integral is $0$.