Chapter 6.1, Problem 33E

(a)

To determine

To show that: The integral function ${\displaystyle {\int }_0^{\frac{\pi }{2}}{\sin }^{n}xdx}=\frac{n-1}{n}{\displaystyle {\int }_0^{\frac{\pi }{2}}{\sin }^{n-2}xdx}$ using the reduction formula.

(b)

To determine

To evaluate: The integral function ${\displaystyle {\int }_0^{\frac{\pi }{2}}{\sin }^{3}xdx}$ and ${\displaystyle {\int }_0^{\frac{\pi }{2}}{\sin }^{5}xdx}$.

(c)

To determine

To show that: The value of the integral function ${\displaystyle {\int }_0^{\frac{\pi }{2}}{\sin }^{2n+1}xdx}$ is $\frac{2\cdot 4\cdot 6\cdot \cdot \cdot 2n}{3\cdot 5\cdot 7\cdot \cdot \cdot \left(2n+1\right)}$ for odd powers of sine.