$\int\limits_{-3 \pi / 2}^{-\pi / 2}\left\{(x+\pi)^3+\cos ^2(x+3 \pi)\right\} d x$, is equal to

$\frac{\pi}{3}$

Putting $x+\pi=t$, we have

$I=\int\limits_{-3 \pi / 2}^{-\pi / 2}\left\{(x+\pi)^3+\cos ^2(x+3 \pi)\right\} d x$

$\Rightarrow I=\int\limits_{-\pi / 2}^{\pi / 2}\left\{t^3+\cos ^2(2 \pi+t)\right\} d t$

$\Rightarrow I=\int\limits_{-\pi / 2}^{\pi / 2} t^3 d t+\int\limits_{-\pi / 2}^{\pi / 2} \cos ^2 t d t=0+2 \int\limits_0^{\pi / 2} \cos ^2 t d t=2 \times \frac{\pi}{4}=\frac{\pi}{2}$