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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 {\rm{/4}}}^{\pi {\rm{/2}}} {{e^x}(\log \sin x + \cot x)\,dx = } $
  • A
    ${e^{\pi /4}}\log 2$
  • B
    $ - {e^{\pi /4}}\log 2$
  • ✓
    $\frac{1}{2}{e^{\pi /4}}\log 2$
  • D
    $ - \frac{1}{2}{e^{\pi /4}}\log 2$
✓

Answer

Correct option: C.
$\frac{1}{2}{e^{\pi /4}}\log 2$
c
(c) Let $I = \int_{\pi /4}^{\pi /2} {{e^x}(\log \sin x + \cot x)dx} $

$I = \int_{\pi /4}^{\pi /2} {{e^x}\log \sin x\,dx + \int_{\pi /4}^{\pi /2} {{e^x}\cot x\,dx} } $

$ = \int_{\pi /4}^{\pi /2} {{e^x}\log \sin xdx + [{e^x}\log \sin x]_{\pi /4}^{\pi /2}} $$ - \int_{\pi /4}^{\pi /2} {{e^x}\log \sin x\,dx} $

$ = {e^{\pi /2}}\log \sin \frac{\pi }{2} - {e^{\pi /4}}\log \sin \frac{\pi }{4} $

$= \frac{1}{2}{e^{\pi /4}}\log 2$.

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