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Definite Integration — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDefinite Integration2 Marks
MCQ
$\int_{\pi / 3}^{\pi / 2} \frac{\sqrt{1+\cos x}}{(1-\cos x)^{\frac{5}{2}}} d x=$
  • A
    $\frac{5}{2}$
  • ✓
    $\frac{3}{2}$
  • C
    $\frac{1}{2}$
  • D
    $\frac{2}{5}$
✓

Answer

Correct option: B.
$\frac{3}{2}$
(B)
Let $I =\int_{\pi / 3}^{\pi / 2} \frac{\sqrt{1+\cos x}}{(1-\cos x)^{\frac{5}{2}}} d x$
$=\int_{\pi / 3}^{\pi / 2} \frac{\sqrt{1+\cos x}}{(1-\cos x)^{\frac{5}{2}}} \times \frac{\sqrt{1-\cos x}}{\sqrt{1-\cos x}} d x$
$=\int_{\pi / 3}^{\pi / 2} \frac{\sin x}{(1-\cos x)^3} d x$
$\begin{array}{l}\text { Put } 1-\cos x= t \\ \Rightarrow \sin x d x= dt \end{array}$
$\therefore \quad I=\int_{1 / 2}^1 \frac{ dt }{ t ^3}=\left[\frac{ t ^{-2}}{-2}\right]_{1 / 2}^1=\frac{3}{2}$

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