A225122 Numerators of Integral_{x=0..Pi/2} sin(2*n*x)*log(cosec(x)) dx.

0, 1, 1, 7, 1, 43, 23, 337, 22, 1091, 563, 12701, 1627, 172673, 88069, 179141, 5692, 3141493, 1593269, 61309987, 7759469, 8966941, 31730711, 1474162241, 93044486, 7510486741, 3788707301, 22918154531, 2888008157, 675037481999, 340028535787

Offset

0,4

Comments

The integral with sin(2*n*x) is always rational, but the integral with sin((2*n+1)*x) is irrational and is of the form p/q+log(2)/(2*n+1).

References

George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 218.

Links

Table of n, a(n) for n=0..30.

Example

0, 1/2, 1/2, 7/18, 1/3, 43/150, 23/90, 337/1470, 22/105, 1091/5670, 563/3150, ... = A225122/A225123

Maple

a:= n-> numer(Re(int(sin(2*n*x)*log(csc(x)), x=0..Pi/2))):
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 29 2013

Mathematica

a[n_] := Integrate[Sin[2*n*x]*Log[Csc[x]], {x, 0, Pi/2}]; Table[a[n] // Numerator, {n, 0, 30}]

Crossrefs

Cf. A225123 (denominators).

Sequence in context: A050402 A027643 A373036 * A051931 A188728 A264617

Adjacent sequences: A225119 A225120 A225121 * A225123 A225124 A225125

Keyword

nonn,easy,frac

Author

Jean-François Alcover, Apr 29 2013

Status

approved