Question

Evaluate: $\underset{0}{\overset{\pi /2}{\int }}\sqrt{cos\theta }si{n}^3\theta d\theta$

• 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$\theta =\frac{\pi }{2},t=0$

$\therefore \underset{0}{\overset{\pi /2}{\int }}\sqrt{cos\theta }si{n}^3\theta d\theta =\underset{0}{\overset{\pi /2}{\int }}\sqrt{cos\theta }\cdot \left(1-co{s}^2\theta \right)\cdot sin\theta d\theta$

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

$=-{\left[\frac{2}{3}{t}^{3/2}-\frac{2}{7}{t}^{7/2}\right]}_1^0$
$=-\left[0-\left(\frac{2}{3}-\frac{2}{7}\right)\right]=\frac{8}{21}$