_0^ 2 e^ x/2 . ( x 2 + 4 ) ,dx = - Vidyadip

MCQ

$\int_0^{2\pi } {{e^{x/2}}.\sin \left( {\frac{x}{2} + \frac{\pi }{4}} \right)\,dx = } $

A
$1$
B
$2\sqrt 2 $
✓
$0$
D
None of these

✓

Answer

Correct option: C.

$0$

c
(c) Let $I = \int_0^{2\pi } {{e^{x/2}}\sin \left( {\frac{x}{2} + \frac{\pi }{4}} \right)\,dx} $

==> $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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