Question

Solve: $\int \sqrt{\frac{\cos x-{\cos }^{3}x}{1-{\cos }^{3}x}dx}=$

Answer 1

$\int \sqrt{\frac{\cos x-{\cos }^{3}x}{1-{\cos }^{3}x}}dx$

$=\int \sqrt{\frac{\cos x(1-{\cos }^{2}x)}{1-{\cos }^{3}x}}dx$

$=\int \sqrt{\frac{\cos x{\sin }^{2}x}{1-{\cos }^{3}x}}dx$

$=\int \frac{\sin x\sqrt{\cos x}}{\sqrt{1-{\cos }^{3}x}}dx$

$\cos x=t$

$-\sin xdx=dt$

$\Rightarrow -\int \frac{\sqrt{t}}{\sqrt{1-t^3}}dt$

$=-\int \frac{\sqrt{t}}{\sqrt{(1{)}^2-(t^{3/2}{)}^2}}dt\times \frac{3}{2}\times \frac{2}{3}$

$\Rightarrow \frac{-2}{3}\int \frac{\frac{3}{2}\sqrt{t}}{\sqrt{(1{)}^2-(t^{3/2}{)}^2}}dt$ [We know $\int \frac{2x}{\sqrt{1-x^2}}dx-{\cos }^{-1}x$]

$\Rightarrow \frac{-2}{3}{\cos }^{-1}\left(t^{3/2}\right)+C$

$I=\frac{-2}{3}{\cos }^{-1}\left({\cos }^{\frac{3}{2}}x\right)+C$.