Question

Value of ${\int }_{-\pi /2}^{\pi /2}\sqrt{\cos x-{\cos }^{3}x}$ dx is

• A. $\frac{2}{3}$
• B. $\frac{4}{3}$
• C. $\frac{5}{3}$
• D. $\frac{1}{3}$

Answer 1

The correct option is B $\frac{4}{3}$

$I={\int }_{-\pi /2}^{\pi /2}\sqrt{cosx(1-co{s}^{2}x)}dx$
$={\int }_{-\pi /2}^{\pi /2}\sqrt{cosx}|sinx|dx$
$=2{\int }_0^{\pi /2}\sqrt{\cos x}\left(\sin x\right)dx$
Put $\cos x=t$
$\Rightarrow -\sin xdx=dt$
So,$I=2{\int }_0^1\sqrt{t}dt$
$=2\left(\frac{2}{3}\right){\left[t^{3/2}\right]}_0^1$
$I=\frac{4}{3}$