nice-calculus-0-pi-2-x-3-cot-x-dx-a-16-a-m-n-

Question Number 151954 by mnjuly1970 last updated on 24/Aug/21

n i c e \dots c a l c u l u s

ϕ := \int_{0}^{\frac{π}{2}} x^{3} . c o t (x) d x = \frac{a}{16}

a := ?

m . n \dots

Answered by Kamel last updated on 24/Aug/21

Ω = \int_{0}^{\frac{π}{2}} x^{3} c o t (x) d x \overset{I B P}{=} - 3 \int_{0}^{\frac{π}{2}} x^{2} L n (s i n (x)) d x

L n (s i n (x)) = - L n (2) - \underset{n = 1}{\sum^{+ \infty}} \frac{c o s (2 n x)}{n}, 0 < x ⩽ \frac{π}{2} .

∴ Ω = \frac{π^{3}}{8} L n (2) + 3 \underset{n = 1}{\sum^{+ \infty}} \frac{1}{n} \int_{0}^{\frac{π}{2}} x^{2} c o s (2 n x) d x

= \frac{π^{3}}{8} L n (2) + \frac{3 π}{2} \underset{n = 1}{\sum^{+ \infty}} \frac{1}{n} (\frac{{(- 1)}^{n}}{2 n^{2}})

= \frac{π^{3}}{8} L n (2) + \frac{3 π}{4} (\frac{1}{8} ζ (3) - (ζ (3) - \frac{1}{8} ζ (3)))

= \frac{π^{3}}{8} L n (2) - \frac{9 π}{16} = \frac{π^{3} L n (4) - 9 π}{16}

Commented by mnjuly1970 last updated on 25/Aug/21

e x c e l l e n t \dots