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7.2 definite integral — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths7.2 definite integral1 Mark
MCQ
$\int_0^{\pi /2} {\frac{{x + \sin x}}{{1 + \cos x}}\,dx = } $
  • A
    $ - \log 2$
  • B
    $\log 2$
  • ✓
    $\frac{\pi }{2}$
  • D
    $0$
✓

Answer

Correct option: C.
$\frac{\pi }{2}$
c
(c) $\int_0^{\pi /2} {\frac{{x + \sin x}}{{1 + \cos x}}dx = \int_0^{\pi /2} {\frac{{x + \sin x}}{{2{{\cos }^2}\frac{x}{2}}}dx} } $

$ = \frac{1}{2}\int_0^{\pi /2} {x{{\sec }^2}\frac{x}{2}} dx + \int_0^{\pi /2} {\tan \frac{x}{2}dx} $.

$ = \left| {\,x\tan \frac{x}{2}\,} \right|_0^{\pi /2} = \frac{\pi }{2}\tan \frac{\pi }{4} = \frac{\pi }{2}$.

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