Question

The value of $3{\int }_0^{\pi /2}\sqrt{\cos x-{\cos }^{3}x}dx$ is

Answer 1

Let $I=\int \sqrt{\cos x-{\cos }^{3}x}dx=\int \sqrt{\cos x{\sin }^{2}x}dx=\int \sin x\sqrt{\cos x}dx$
Substituting $u=\cos x\Rightarrow du=-\sin xdx$
$I=-\int \sqrt{u}du=-\frac{2u^{2/3}}{3}+c=-\frac{2{\cos x}^{2/3}}{3}+c$
Therefore, $3{\int }_0^{\pi /2}\sqrt{\cos x-{\cos }^{3}x}dx=2$