Answer
$$ \frac{-1}{499}\cos^{499}y+\frac{1}{501}\cos^{501}+c$$
Work Step by Step
\begin{align*}
\int \cos ^{498} y \sin ^{3} y d y&=\int \cos ^{498} y \sin ^{2} y\sin y d y\\
&=\int \cos^{498}y(1-\cos^2 y)\sin ydy\\
&= \int (\cos^{498}y-\cos^{500} y)\sin ydy\\
&=\frac{-1}{499}\cos^{499}y+\frac{1}{501}\cos^{501}+c
\end{align*}
