The value of $\int\limits_{-\pi / 6}^{\pi / 6}\left(4 x^5+x \sin ^2 x+\tan ^3 x+2\right) dx$ is:

$\frac{2 \pi}{3}$

The correct answer is Option (1) → $\frac{2 \pi}{3}$

$I=\int\limits_{-\pi / 6}^{\pi / 6}\left(4 x^5+x \sin ^2 x+\tan ^3 x+2\right) dx$

$4 x^5+x \sin ^2 x+\tan ^3 x$ → odd function

2 → even function

$I=\int\limits_{-\pi / 6}^{\pi / 6}0+2dx$

$=2×\frac{\pi}{3}=\frac{2 \pi}{3}$