\begin{align} I &= \int_0^\pi e^x \sin(x) \cos(x) dx \\ &= \frac{1}{2} \int_0^\pi e^x \sin(2x) dx \\ &= \frac{1}{2} \left[ e^x \sin(2x) \right]_0^\pi - \int_0^\pi e^x \cos(2x) dx \\ &= - \left[ e^x \cos(2x) \right]_0^\pi - 2 \int_0^\pi e^x \sin(2x) dx \\ &= - e^\pi + 1 - 4I \\ \therefore \ \ 5 I &= 1 - e^\pi \\ \therefore \ \ I &= \frac{1 - e^\pi}{5} \end{align}