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STD 12 - 7.2 definite integral — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 7.2 definite integral1 Mark
MCQ
$\int_0^1 {\log \sin \left( {\frac{\pi }{2}x} \right)} \,dx = $
  • ✓
    $ - \log 2$
  • B
    $\log 2$
  • C
    $\frac{\pi }{2}\log 2$
  • D
    $ - \frac{\pi }{2}\log 2$
✓

Answer

Correct option: A.
$ - \log 2$
a
(a) Put $\frac{\pi }{2}x = \theta \Rightarrow dx = \frac{2}{\pi }d\theta $; 

As $x = 0$ to $1,$ $\theta = 0$ to $\frac{\pi }{2}$

Then it reduces to 

$\frac{2}{\pi }\int_0^{\pi /2} {\,\,\log \sin \theta \,d\theta = \frac{2}{\pi }\left[ { - \frac{\pi }{2}\log 2} \right]} $

$ = - \log 2$.

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