$\int\limits_{0}^{2π}e^{x/2}\sin(\frac{x}{2}+\frac{π}{4})dx=$

$2π$ $e^π$ 0 $2\sqrt{2}$

0

We have, $I=\int\limits_{0}^{2π}e^{x/2}\sin(\frac{x}{2}+\frac{π}{4})dx$ $⇒I=2\int\limits_{0}^{π}e^{t}\sin(\frac{π}{4}+t)dt$, where $x=2t$ $⇒I=\frac{2}{\sqrt{2}}\int\limits_{0}^{π}e^t(\sin t+\cos t)dt=\sqrt{2}\left[e^t\,\sin t\right]_{0}^{π}=\sqrt{2}×0=0$