Chapter B.4, Problem 14P

(a)

To determine

Whether ${\displaystyle \underset{0}{\overset{\pi }{\int }}{\left(\sin \left(\theta \right)\right)}^{n}d\theta =\frac{\sqrt{\pi }\Gamma \left(\frac{n}{2}+\frac{1}{2}\right)}{\Gamma \left(\frac{n}{2}+1\right)}}$ satisfy for $n=0,1$ .

(b)

To determine

The proof of the equation ${\displaystyle \underset{0}{\overset{\pi }{\int }}{\left(\sin \theta \right)}^{n}d\theta =\frac{n-1}{n}}{\displaystyle \underset{0}{\overset{\pi }{\int }}{\left(\sin \theta \right)}^{n-2}d\theta }$ .

(c)

To determine

The proof of the equation ${\displaystyle \underset{0}{\overset{\pi }{\int }}{\left(\sin \left(\theta \right)\right)}^{n}d\theta =\frac{\sqrt{\pi }\Gamma \left(\frac{n}{2}+\frac{1}{2}\right)}{\Gamma \left(\frac{n}{2}+1\right)}}$ .