Answer
$$-\frac{1}{3}\cos^3 x+\frac{2}{5}\cos^5x-\frac{1}{7}\cos^7x+C$$
Work Step by Step
\begin{align*}
\int \sin ^{5} x \cos ^{2} x d x&=\int \sin ^{4} x \cos ^{2} x \sin xd x\\ &=\int (1-\cos^{2}x)^2 \cos ^{2} x \sin xd x\\
&=\int ( \cos ^{2} x-2\cos^{4}x+\cos^6x)\sin xdx\\
&=-\frac{1}{3}\cos^3 x+\frac{2}{5}\cos^5x-\frac{1}{7}\cos^7x+C
\end{align*}
