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STD 12 - 7.2 definite integral — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 7.2 definite integral4 Marks
MCQ
$\int\limits_0^{\frac{\pi }{2}} {\,\,\sqrt {\sin \,2\theta } } \sin\, \theta \,d\theta$ is equal to :
  • A
    $0$
  • ✓
    $\pi /4$
  • C
    $\pi /2$
  • D
    $\pi$
✓

Answer

Correct option: B.
$\pi /4$
b
$I =\int\limits_0^{\frac{\pi }{2}} {\,\,\sqrt {\sin \,2\theta } }$$.\cos\, \theta \,d\theta$ $\Rightarrow 2 \,I = \int\limits_0^{\frac{\pi }{2}} {\,\,\sqrt {\sin \,2\theta } } (\cos\, \theta + \sin\, \theta )\, d\theta$
$= \int\limits_0^{\frac{\pi }{2}} {\,\,\sqrt {1\,\, - \,\,{{\left( {\sin \,\theta \,\, - \,\,\cos \,\theta } \right)}^2}} }  . (\cos\, \theta + \sin \,\theta )\, d\theta$ Put $\sin \,\theta - \cos\, \theta = t$

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