Question

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

Answer 1

Answer: 1.41

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