Definite Integration Question 360
Question: $ \int_0^{2\pi }{{e^{x/2}}.\sin ( \frac{x}{2}+\frac{\pi }{4} )dx=} $
[Roorkee 1982]
Options:
A) 1
B) $ 2\sqrt{2} $
C) 0
D) None of these
Show Answer
Answer:
Correct Answer: C
Solution:
Let $ I=\int_0^{2\pi }{{e^{x/2}}\sin ( \frac{x}{2}+\frac{\pi }{4} )dx} $
therefore $ I=2\int_0^{\pi }{e^{t}\sin ( t+\frac{\pi }{4} )dt}=2[ \frac{e^{t}}{\sqrt{1+1}}\sin ( t+\frac{\pi }{4}-{{\tan }^{-1}}\frac{1}{1} ) ]_0^{\pi } $
$ =\frac{2}{\sqrt{2}}[ e^{t}\sin t ]_0^{\pi }=\frac{2}{\sqrt{2}}[0]=0 $ .
BETA
