Answer
$$
\int \sinh x \cosh ^{2} x d x =\frac{1}{3} \cosh ^{3} x+C
$$
where $C$ is an arbitrary constant.
Work Step by Step
$$
\int \sinh x \cosh ^{2} x d x
$$
Let $u=\cosh x . $ Then $d u=\sinh x d x, $. and, the Substitution Rule gives
$$
\begin{aligned}
\int \sinh x \cosh ^{2} x d x &=\int u^{2} d u \\
&=\frac{1}{3} u^{3}+C \\
&=\frac{1}{3} \cosh ^{3} x+C
\end{aligned}
$$
where $C$ is an arbitrary constant.