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7.2 definite integral — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths7.2 definite integral1 Mark
MCQ
$\int_{ - \pi /4}^{\pi /2} {{e^{ - x}}\sin x\,dx} = $
  • ✓
    $ - \frac{1}{2}{e^{ - \pi /2}}$
  • B
    $ - \frac{{\sqrt 2 }}{2}{e^{ - \pi /4}}$
  • C
    $ - \sqrt 2 ({e^{ - \pi /4}} + {e^{ - \pi /4}})$
  • D
    $0$
✓

Answer

Correct option: A.
$ - \frac{1}{2}{e^{ - \pi /2}}$
a
(a) $\int_{ - \pi /4}^{\pi /2} {{e^{ - x}}\sin x\,dx} $

$= \left[ {\frac{{{e^{ - x}}}}{2}( - \sin x - \cos x)} \right]_{ - \pi /4}^{\pi /2}$

$ = \frac{1}{2}[{e^{ - x}}( - \sin x - \cos x)]_{ - \pi /4}^{\pi /2}$

$ = \frac{1}{2}\left[ {{e^{ - \pi /2}}( - 1 - 0) - \left\{ {{e^{\pi /4}}\left( {\frac{1}{{\sqrt 2 }} - \frac{1}{{\sqrt 2 }}} \right)} \right\}} \right] $

$= - \frac{{{e^{ - \pi /2}}}}{2}$.

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