Question

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

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

Answer 1

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

$\int e^{x/2}\sin (\frac{\pi }{4}+\frac{x}{z})dx$

$=\frac{1}{D}{e}^{x/2}\sin (\frac{\pi }{4}+\frac{x}{z})$

$=e^{x/2}\frac{1}{2+D}\sin (\frac{\pi }{4}+\frac{x}{z})=e^{x/2}\left(\frac{2-D}{4-D^2}\right)\sin (\frac{\pi }{4}+\frac{x}{z})dx$

$\frac{e^{x/2}2\sin (\frac{\pi }{4}+\frac{x}{2})-\cos (\frac{\pi }{4}+\frac{x}{2})}{4-(-1)}+C$

$\frac{e^{x/2}2\sin (\frac{\pi }{4}+\frac{x}{2})-\cos (\frac{\pi }{4}+\frac{x}{2})}{5}+C$

after using $\sin (A+B)$ and $\cos (A-B)$

$=\sqrt{2}{e}^{x/2}\sin \left(\frac{x}{2}\right)+C$