Question

${\int }_0^{\frac{x}{2}}\sqrt{cos\theta }si{n}^3\theta d\theta =$

• A. $\frac{20}{21}$
• B. $\frac{8}{21}$
• C. $\frac{-20}{21}$
• D. $\frac{-8}{21}$

Answer 1

The correct option is B $\frac{8}{21}$

$LetI={\int }_0^{\frac{\pi }{2}}\sqrt{cos\theta }si{n}^3\theta d\theta$
$Putt=cos\theta \Rightarrow dt=-sin\theta d\theta ,then$
$I=-{\int }_1^0{t}^{\frac{1}{2}}(1-t^2)dt={\int }_0^1(t^{\frac{1}{2}}-t^{\frac{5}{2}})dt$
$I={[\frac{2}{3}{t}^{\frac{3}{2}}-\frac{2}{7}{t}^{\frac{7}{2}}]}_0^1=\frac{8}{21}$