Answer
$$\frac{e^4+3}{2e^2}$$
Work Step by Step
Given $$ \int_{0}^{2}y\sinh ydy$$
Let
\begin{align*}
u&=y\ \ \ \ \ \ \ \ \ \ \ \ \ \ dv=\sinh ydy\\
u&= dy\ \ \ \ \ \ \ \ \ \ \ \ \ dv=\cosh y
\end{align*}
Then using integration by parts
\begin{align*}
\int_{0}^{2} y\sinh y dy &=uv-\int vdu\\
&= y\cosh y\bigg|_{0}^{2}-\int_{0}^{2} \cosh ydy\\
&=y\cosh y-\sinh y \bigg|_{0}^{2}\\
&=\frac{e^4+3}{2e^2}
\end{align*}