Answer
$$\frac{4}{e}$$
Work Step by Step
Given
$$ \int_{0}^{\pi} e^{\cos t}\sin 2t dt$$
Let $ s=\cos t \ \ \Rightarrow \ ds=\sin t dt $, at $t=0\to s=1$at $t=\pi\to s=-1$
$$ \int_{0}^{\pi} e^{\cos t}\sin 2t dt=-2 \int_{-1}^{1}se^sds$$
Let
\begin{align*}
u&=s\ \ \ \ \ \ \ \ \ \ \ \ \ \ dv=e^sds\\
u&= ds\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ v=e^s
\end{align*}
then
\begin{align*}
\int_{0}^{\pi} e^{\cos t}\sin 2t dt&=-2 \int_{-1}^{1}se^sds\\
&=-2\left( se^s\bigg|_{-1}^{1}-\int_{-1}^{1} e^sds\right)\\
&=-2\left( se^s- e^s \right)\bigg|_{-1}^{1}\\
&=\frac{4}{e}
\end{align*}
