Question

If $\int e^{x / 2} \sin \left(\frac{x}{2}+\frac{\pi}{4}\right) d x=k e^{x / 2} \sin \frac{x}{2}+c$ then find $k$

Answer 1

$\int e ^{\frac{ x }{2}} \sin \left(\frac{ x }{2}+\frac{\pi}{4}\right) dx$
$=\int e ^{\frac{ x }{2}} \frac{1}{\sqrt{2}}\left(\sin \frac{ x }{2}+\cos \frac{ x }{2}\right) dx$
Let $\frac{x}{2}= t \Rightarrow dx =2 dt$
$=\sqrt{2} \int e^{t}(\sin t+\cos t) d t$
$\because \frac{d}{d t}(\sin t)=\cos t$
$I=\sqrt{2} e^{t} \sin t$
$=\sqrt{2} e^{\frac{x}{2}} \sin \frac{x}{2}+c$