I-0-pi-x-pi-x-sin-x-dx-

Question Number 1085 by 123456 last updated on 10/Jun/15

I = \underset{0}{\int^{π}} \frac{x (π - x)}{\sin x} d x

Commented by 123456 last updated on 10/Jun/15

f (x) = \frac{x (π - x)}{\sin x}

f (0^{+}) \overset{?}{=} π

f (π^{-}) \overset{?}{=} π

\exists ξ \in (0, π), \forall x \in (0, π), f (ξ) ⩾ f (x)

\exists ζ \in (0, π), \forall x \in (0, π), f (ζ) ⩽ f (x)

π f (ζ) \overset{?}{⩽} \underset{0}{\int^{π}} f (x) d x \overset{?}{⩽} π f (ξ)

Commented by 123456 last updated on 10/Jun/15

f (\frac{π}{2}) \overset{?}{=} \frac{π^{2}}{2 \sqrt{2}}

π^{2} \overset{?}{<} \underset{0}{\int^{π}} f (x) d x \overset{?}{<} \frac{π^{3}}{2 \sqrt{2}}

Commented by prakash jain last updated on 10/Jun/15

\lim_{x \to 0} \frac{x (π - x)}{\sin x} = π

\lim_{x \to π} \frac{x (π - x)}{\sin x} = π