Question

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

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

Answer 1

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

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