$\int\limits_{0}^{2π}e^{x/2}\sin(\frac{x}{2}+\frac{π}{4})dx=$
Answer & explanation
Correct answer: option 3
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 $x=2t$
$⇒I=\frac{2}{\sqrt{2}}\int\limits_{0}^{π}e^t(\sin t+\cos t)dt=\sqrt{2}\left[e^t\,\sin t\right]_{0}^{π}=\sqrt{2}×0=0$