0-pi-2-x-3-sin-2-x-dx-3-8-pi-2-ln-4-7-3-

Question Number 165380 by mnjuly1970 last updated on 31/Jan/22

\int_{0}^{\frac{π}{2}} \frac{x^{3}}{{s i n}^{2} (x)} d x \overset{?}{=} \frac{3}{8} (π^{2} l n (4) - 7 ζ (3))

Answered by Ar Brandon last updated on 31/Jan/22

I = \int_{0}^{\frac{π}{2}} x^{3} {cosec}^{2} x d x = - {[x^{3} \cot x]}_{0}^{\frac{π}{2}} + 3 \int_{0}^{\frac{π}{2}} x^{2} \cot x d x

= 3 {[x^{2} \ln (\sin x)]}_{0}^{\frac{π}{2}} - 6 \int_{0}^{\frac{π}{2}} x \ln (\sin (x) d x