Question

Evaluate: $ \int\limits_{0}^{\pi/2} \sqrt{cos\,\theta}\, sin^{3} \theta \, d\theta$

• A. $\frac{8}{21}$
• B. $\frac{7}{21}$
• C. $\frac{8}{23}$
• D. $\frac{7}{23}$

Answer 1

Answer: (A)

Let $ cos \,\theta =t$
$ \Rightarrow -sin\,\theta \,d\theta = dt $

when$\, \theta = 0, t = 1\,\,$ and$\,\, \theta = \frac{\pi}{2 }, t = 0 $

$\therefore \int\limits_{0}^{\pi/2} \sqrt{cos\,\theta}\, sin^{3} \,\theta \,d\theta = \int\limits_{0}^{\pi /2} \sqrt{cos\,\theta }\cdot\left(1-cos ^{2}\theta\right)\cdot sin\, \theta \,d\theta $

$= - \int\limits_{0}^{1} \sqrt{t}\left(1-t^{2}\right)dt = - \int\limits_{1}^{0} \left(\sqrt{t} -t^{5/2}\right) dt$

$ = -\left[\frac{2}{3} t^{{3}/{2}}-\frac{2}{7} t^{{7}/{2}}\right]_{1}^{0}$
$ = -\left[0 - \left( \frac{2}{3} -\frac{2}{7}\right)\right]= \frac{8}{21}$