Question

$\int e^{x/2}\sin \left(\frac{x}{2}+\frac{\pi }{4}\right)dx$ is equal to.

• A. $e^{x/2}\cos \frac{x}{2}+C$
• B. $\sqrt{2}{e}^{x/2}\cos \frac{x}{2}+C$
• C. $e^{x/2}\sin \frac{x}{2}+C$
• D. $\sqrt{2}{e}^{x/2}\sin \frac{x}{2}+C$

Answer 1

The correct option is D $\sqrt{2}{e}^{x/2}\sin \frac{x}{2}+C$

Let $I=\int e^{x/2}\cdot \sin \left(\frac{x}{2}+\frac{\pi }{4}\right)dx$
$=2\sin \left(\frac{x}{2}+\frac{\pi }{4}\right)e^{x/2}-\int \cos \left(\frac{x}{2}+\frac{\pi }{4}\right)\cdot \frac{1}{2}\cdot 2e^{x/2}dx+C$
$=2\sin \left(\frac{x}{2}+\frac{\pi }{4}\right)\cdot e^{x/2}-2e^{x/2}\cdot \cos \left(\frac{x}{2}+\frac{\pi }{4}\right)-\int \sin \left(\frac{x}{2}+\frac{\pi }{4}\right)\cdot \frac{1}{2}\cdot 2e^{x/2}dx$
Therefore,
$2I=2e^{x/2}\{\sin \left(\frac{x}{2}+\frac{\pi }{4}\right)-\cos \left(\frac{x}{2}-\frac{\pi }{4}\right)\}$
$\Rightarrow I=e^{x/2}\{\sin \left(\frac{x}{2}+\frac{\pi }{4}\right)-\cos \left(\frac{x}{2}+\frac{\pi }{4}\right)\}$
$=\sqrt{2}{e}^{x/2}\cdot \left(\sin \frac{x}{2}\right)=\sqrt{2}\cdot e^{x/2}\cdot \sin \frac{x}{2}+C$.
Trick: By inspection,
$\frac{d}{dx}[\sqrt{2}{e}^{x/2}\cdot \sin \frac{x}{2}+C]$
$=\sqrt{2}\left[\frac{1}{2}{e}^{x/2}\cdot \cos \frac{x}{2}+\frac{1}{2}{e}^{x/2}\cdot \sin \frac{x}{2}\right]$
$=e^{x/2}\left[\frac{1}{\sqrt{2}}\cos \frac{x}{2}+\frac{1}{\sqrt{2}}\sin \frac{x}{2}\right]$
$=e^{x/2}\cdot \sin \left(\frac{x}{2}+\frac{\pi }{4}\right)$.