Answer
$$-\frac{1}{3} \cos ^{3} \theta+\frac{1}{5} \cos ^{5} \theta+c $$
Work Step by Step
\begin{aligned} \int \sin ^{3} \theta \cos ^{2} \theta d \theta &=\int \sin ^{2} \theta \cos ^{2} \theta \sin \theta d \theta \\ &=\int\left(1-\cos ^{2} \theta\right) \sin \theta \cos ^{2} \theta d \theta \\ &=\int\left(\cos ^{2} \theta-\cos ^{4} \theta\right) \sin \theta d \theta \\ &=\int\left(\cos ^{2} \theta\right) \sin \theta d \theta-\int\left(\cos ^{4} \theta\right) \sin \theta d \theta \\ &=-\frac{1}{3} \cos ^{3} \theta+\frac{1}{5} \cos ^{5} \theta+c \end{aligned}
