First slide

Methods of integration

Question

$\int \sqrt{\frac{\cos x - \cos^{3} x}{1 - \cos^{3} x}} d x is equal to$

Easy

A

$\frac{2}{3} \sin^{- 1} (\cos^{3 / 2} x) + C$
B

$\frac{3}{2} \sin^{- 1} (\cos^{3 / 2} x) + C$
C

$\frac{2}{3} \cos^{- 1} (\cos^{3 / 2} x) + C$
D

$None of these$

Solution

$\begin{array}{l} \int \sqrt{\frac{\cos x - \cos^{3} x}{1 - \cos^{3} x}} d x = \int \sqrt{\frac{\cos x}{1 - \cos^{3} x}} \sin x d x \\ = \int \sqrt{\frac{t}{1 - t^{3}}} d t = - \int \frac{\sqrt{t}}{\sqrt{1 - {(t^{3 / 2})}^{2}}} d t, where t = \cos x \\ = - \frac{2}{3} \int \frac{\frac{3}{2} \sqrt{t}}{\sqrt{1 - {(t^{3 / 2})}^{2}}} d t = \frac{2}{3} \cos^{- 1} (t^{3 / 2}) + C \\ = \frac{2}{3} \cos^{- 1} (\cos x)^{3 / 2} + C \end{array}$

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