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