A225125 Decimal expansion of Integral_{x=0..Pi/2} x^3*cosec(x) dx.

1, 6, 9, 2, 9, 9, 2, 4, 6, 8, 4, 1, 3, 6, 0, 1, 2, 4, 4, 6, 7, 8, 0, 1, 3, 8, 3, 4, 8, 9, 8, 1, 0, 8, 7, 0, 8, 0, 7, 8, 6, 9, 8, 6, 7, 1, 5, 6, 8, 0, 7, 2, 3, 4, 9, 5, 6, 8, 8, 0, 1, 5, 7, 7, 8, 9, 4, 7, 6, 4, 3, 7, 2, 1, 3, 1, 9, 8, 7, 9, 8, 7, 2, 7, 9, 1, 8, 7, 3, 6, 3, 9, 6, 3, 5, 4, 4, 9, 4, 2

Offset: 1

Author

Jean-François Alcover, Apr 30 2013

Keywords

• nonn
• cons
• easy

Comments

The simpler Integral_{x=0..Pi/2} x*cosec(x) dx evaluates as 2*Catalan.

Examples

			1.6929924684136012446780138348981087080786986715680723495688...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• StackExchange, An integral with PolyGamma.

Crossrefs

Cf. A006752, A221209.

Programs

Mathematica

3*Catalan*Pi^2/2-1/128*(PolyGamma[3, 1/4]-PolyGamma[3, 3/4]); (* or *)
3*Catalan*Pi^2/2-3/64*(Zeta[4, 1/4]-Zeta[4, 3/4]) // RealDigits[#, 10, 100] & // First
RealDigits[Integrate[x^3 Csc[x],{x,0,Pi/2}],10,120][[1]] (* Harvey P. Dale, Jun 16 2025 *)

PARI

3*Catalan*Pi^2/2-3/64*(zetahurwitz(4,1/4)-zetahurwitz(4,3/4)) \\ Charles R Greathouse IV, Jan 31 2018

Formula

Equals 3*Catalan*Pi^2/2-1/128*(polygamma(3, 1/4)-polygamma(3, 3/4)).