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