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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_{\,\pi /6}^{\,\pi /3} {\,\frac{{dx}}{{1 + \sqrt {\cot x} }}} $ is
  • A
    $\pi /3$
  • B
    $\pi /6$
  • ✓
    $\pi /12$
  • D
    $\pi /2$
✓

Answer

Correct option: C.
$\pi /12$
c
(c) $I = \int_{\pi /6}^{\pi /3} {\frac{{dx}}{{1 + \sqrt {\cot x} }} = } \int_{\pi /6}^{\pi /3} {\frac{{\sqrt {\sin x} }}{{\sqrt {\sin x} + \sqrt {\cos x} }}\,} dx$ ....(i)

$I = \int_{\pi /6}^{\pi /3} {\frac{{\sqrt {\cos x} }}{{\sqrt {\cos x} + \sqrt {\sin x} }}\,} dx$ .....(ii)

Adding (i) and (ii), 

$2I = \int_{\,\pi /6}^{\,\pi /3} {dx} $; 

$I = \frac{1}{2}\left( {\frac{\pi }{3} - \frac{\pi }{6}} \right) = \frac{\pi }{{12}}$.

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