$\underset{0}{\overset{2\pi }{\int }}{e}^{x/2}\sin (\frac{x}{2}+\frac{\pi }{4})dx=$ 

$\begin{array}{c}\frac{x}{2}=t\\ dx=2dt\end{array}$

${\displaystyle \underset{0}{\overset{2\pi }{\int }}{e}^{x/2}\left(Sin\left(\frac{x}{2}+\frac{\pi }{4}\right)\right)dx}$ $=2{\displaystyle \underset{0}{\overset{\pi }{\int }}{e}^t\left(\frac{1}{\sqrt{2}}S\mathrm{in}t+\frac{1}{\sqrt{2}}Cost\right)dt}$

                                                $=\sqrt{2}{\left(e^{t}Sint\right)}_0^{\pi }$

                                                 $=0$