Answer
$2\cosh 2-\sinh 2$
Work Step by Step
Integration by parts:
$\displaystyle \int udv=uv-\int vdu$
The idea is to choose a relatively easy $dv$ to integrate, and
a u whose $u'$ does not complicate matters (best case: makes thing simpler).
----
$\left[\begin{array}{ll}
u=y & dv=\sinh ydy\\
& \\
du=dy & v=\cosh y
\end{array}\right]$
$I=\displaystyle \int_{0}^{2}y\sinh ydy=uv|_{0}^{2}-\int_{0}^{2}vdu$
$=\displaystyle \left.y\cosh y\right|_{0}^{2}-\int_{0}^{2}\cosh ydy$
$=(2\cosh 2-0)-\left[\sinh y\right]_{0}^{2}$
$=2\cosh 2-\sinh 2$
