Question

Solve ${\int }_{-\pi /2}^{\pi /2}\sqrt{\cos x-{\cos }^{3}x}dx$ equals.

Answer 1

Let $I={\int }_{-\pi /2}^{\pi /2}\sqrt{\cos x-{\cos }^{3}x}dx$

Here, $f\left(x\right)=\sqrt{\cos x-{\cos }^{3}x}$
$f(-x)=\sqrt{\cos (-x)-{\cos }^3(-x)}=\sqrt{\cos x-{\cos }^{3}x}$

$\Rightarrow f(-x)=f\left(x\right)$

$\therefore I=2{\int }_0^{\pi /2}\sqrt{\cos x-{\cos }^{3}x}dx=2{\int }_0^{\pi /2}\sqrt{\cos x}\sin xdx$

Put $\cos x=t$

$\Rightarrow -\sin xdx=dt$

When $x=0\Rightarrow t=1$

When $x=\frac{\pi }{2}\Rightarrow t=0$

$I=2{\int }_0^1\sqrt{t}dt$

$=2\cdot \frac{2}{3}\left[\right(t{)}^{3/2}{]}_0^1=\frac{4}{3}$