We have, $$\begin{aligned} f(x) & =\int_{-\pi}^\pi \sin ^3 x \cos ^2 x d x \\ f(-x) & =\int_{-\pi}^\pi \sin ^3(-2)-\cos ^2(-x) d x \\ & =-f(x) \end{aligned}$$

$$\begin{aligned} &\text { Since, } f(x) \text { is an odd function. }\\ &\therefore \quad \int_{-\pi}^\pi \sin ^3 x \cos ^2 x d x=0 \end{aligned}$$