Question

${\int }_0^{\pi /2}\sqrt{cos\theta }si{n}^3\theta d\theta$

Answer 1

${\int }_0^{\pi /2}\sqrt{\cos \theta }{\sin }^3\theta d\theta$

Let $cos\theta =t$

$-\sin \theta d\theta =dt$

$\int \sqrt{\cos \theta }{\sin }^2\theta \sin \theta d\theta$

$\int \sqrt{\cos \theta }(1-{\cos }^2\theta )\sin \theta d\theta$

$\int \sqrt{t}(1-t^2)(-dt)$

$\int \sqrt{t}(t^2-1)\left(dt\right)$

$\int (t^{5/2}-t^{1/2})dt$

$\frac{2}{7}{t}^{7/2}-\frac{2}{3}{t}^{3/2}$

$\left[\frac{2}{7}\right(\cos \theta {)}^{7/2}-\frac{2}{3}(\cos \theta {)}^{3/2}{]}_0^{\pi /2}$

$\left(\frac{2}{7}\right(\cos \left(\pi /2\right){)}^{7/2}-\frac{2}{3}\left(\cos \right(\pi /2\left){)}^{3/2}\right)-\left(\frac{2}{7}\right(\cos 0{)}^{7/2}-\frac{2}{3}\left(\cos 0{)}^{3/2}\right)$

$=0-(\frac{2}{7}-\frac{2}{3})$

$=\frac{2}{3}-\frac{2}{7}$

$=\frac{8}{21}$