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7.2 definite integral — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths7.2 definite integral1 Mark
MCQ
$\int_{\pi /3}^{\pi /2} {\frac{{\sqrt {1 + \cos x} }}{{{{(1 - \cos x)}^{\frac{5}{2}}}}}} \,dx = $
  • A
    $\frac{5}{2}$
  • ✓
    $\frac{3}{2}$
  • C
    $\frac{1}{2}$
  • D
    $\frac{2}{5}$
✓

Answer

Correct option: B.
$\frac{3}{2}$
b
(b) $I = \int_{\pi /3}^{\pi /2} {\frac{{\sqrt {1 + \cos x} }}{{{{(1 - \cos x)}^{5/2}}}} \times \frac{{\sqrt {1 - \cos x} }}{{\sqrt {1 - \cos x} }}} \,\,dx$

$= \int_{\pi /3}^{\pi /2} {\,\,\frac{{\sin x}}{{{{(1 - \cos x)}^3}}}\,dx} $

Now, put $1 - \cos x = t$

Also, when $x = \frac{\pi }{3},t = \frac{1}{2}$ and $x = \frac{\pi }{2}\,,\,\,t = 1$

Therefore, $I = \int_{1/2}^1 {\frac{{dt}}{{{t^3}}} = \left| {\frac{{{t^{ - 2}}}}{{ - 2}}} \right|} _{1/2}^1 = \frac{3}{2}$.

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