Chapter 8, Problem 125RE

a.

To determine

To show: The value of the integral ${\displaystyle {\int }_0^{\pi }{\sin }^{m}xdx}=\frac{m-1}{m}{\displaystyle {\int }_0^{\pi }{\sin }^{m-2}xdx}$ for any integer $m\ge 2$ by using the reduction formula.

b.

To determine

To show: The value of the integral ${\displaystyle {\int }_0^{\pi }{\sin }^{2n}xdx}=\frac{2n-1}{2n}\cdot \frac{2n-3}{2n-2}\cdots \cdot \frac{1}{2}\cdot \pi$ for any integer $n\ge 1$.

c.

To determine

To show: The value of the integral ${\displaystyle {\int }_0^{\pi }{\sin }^{2n+1}xdx}=\frac{2n}{2n+1}\cdot \frac{2n-2}{2n-1}\cdots \cdot \frac{2}{3}\cdot 2$ for any integer $n\ge 1$.

d.

To determine

To show: The ineqality $0\le {\displaystyle {\int }_0^{\pi }{\sin }^{m+1}x}dx\le {\displaystyle {\int }_0^{\pi }{\sin }^{m}x}dx$ for any integer $m\ge 1$ by using the inequality $0\le \sin x\le 1$ on $\left[0,\pi \right]$.

e.

To determine

To show: The ineqality

$\frac{2n}{2n+1}\cdot \frac{2n-2}{2n-1}\cdots \frac{2}{3}\cdot 2\le \frac{2n-1}{2n}\cdot \frac{2n-3}{2n-2}\cdots \frac{1}{2}\cdot \pi \le \frac{2n-2}{2n-1}\cdot \frac{2n-4}{2n-3}\cdots \cdot \frac{2}{3}\cdot 2$ by using the part (d).

f.

To determine

To show: The ineqality $\frac{2n}{2n+1}\cdot \frac{\pi }{2}\le \frac{{\left(2n\right)}^2}{2n+1}\cdot \frac{{\left(2n-2\right)}^2}{{\left(2n-1\right)}^2}\cdots \frac{2^2}{3^2}\le \frac{\pi }{2}$, for any integer $n\ge 2$.

g.

To determine

To prove: The equation $\frac{\pi }{2}=\underset{x\to \infty }{\lim }\left(\frac{2}{1}\cdot \frac{2}{3}\cdot \frac{4}{3}\cdot \frac{4}{5}\cdots \frac{2n}{2n-1}\cdot \frac{2n}{2n+1}\right)$ by using the part (f).