A) 1
B) \[2\sqrt{2}\]
C) 0
D) None of these
Correct Answer: C
Solution :
Let \[I=\int_{0}^{2\pi }{{{e}^{x/2}}\sin \left( \frac{x}{2}+\frac{\pi }{4} \right)\,dx}\] Þ \[I=2\int_{0}^{\pi }{{{e}^{t}}\sin \left( t+\frac{\pi }{4} \right)dt}=2\left[ \frac{{{e}^{t}}}{\sqrt{1+1}}\sin \left( t+\frac{\pi }{4}-{{\tan }^{-1}}\frac{1}{1} \right) \right]_{0}^{\pi }\] \[=\frac{2}{\sqrt{2}}\left[ {{e}^{t}}\sin t \right]_{0}^{\pi }=\frac{2}{\sqrt{2}}[0]=0\].
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