Chapter 8: Techniques of Integration - Section 8.3 - Trigonometric Integrals - Exercises 8.3 - Page 463: 59

$$-\frac{2}{5} \cos ^{5} \theta+C $$

We integrate as follows: \begin{align*} \int \cos ^{3} \theta \sin 2 \theta d \theta&=\int \cos ^{3} \theta(2 \sin \theta \cos \theta) d \theta\\ &=2 \int \cos ^{4} \theta \sin \theta d \theta\\ &=-\frac{2}{5} \cos ^{5} \theta+C \end{align*}