Question

$\int \sqrt{\frac{cosx-co{s}^{3}x}{1-co{s}^{3}x}}$.dx is equal to

• A. $\frac{2}{3}si{n}^{-1}\left(co{s}^{3/2}x\right)+C$
• B. $\frac{3}{2}si{n}^{-1}\left(co{s}^{3/2}x\right)+C$
• C. $\frac{2}{3}co{s}^{-1}\left(co{s}^{3/2}x\right)+C$
• D. none of these

Answer 1

The correct option is C $\frac{2}{3}co{s}^{-1}\left(co{s}^{3/2}x\right)+C$

$I=\int \sqrt{\frac{cosx(1-co{s}^{2}x)}{1-co{s}^{3}x}}dx$
$=\int \frac{\sqrt{cosx.}sinx}{\sqrt{1-(co{s}^{3/2}x{)}^2}}.dx$
Put $co{s}^{3/2}x=t$
$\Rightarrow \frac{3}{2}\sqrt{cosx}(-sinx).dx=dt$
$\therefore I=\frac{2}{3}\int \frac{-dt}{\sqrt{1-t^2}}$
$=\frac{2}{3}co{s}^{-1}\left(t\right)+C$
$=\frac{2}{3}co{s}^{-1}\left(co{s}^{3/2}x\right)+C$