ABSTRACT
The compilation by I. S. Gradshteyn and I. M. Ryzhik [40] contains about 600 pages of definite integrals. Some of them are quite elementary; for instance, 4.291.1 ∫ 1
ln(1 + x)
x dx =
π2
12 (14.1.1)
is obtained by expanding the integrand as a power series and using the value
(−1)k−1 k2
= π2
12 . (14.1.2)
The latter is reminiscent of the series
k2 =
π2
6 . (14.1.3)
The reader will find in [22] many proofs of the classical evaluation (14.1.3). Most entries in [40] appear quite formidable, and their evaluation requires
a variety of methods and ingenuity. Entry 4.229.7
ln ln tanx dx = π
2 ln
( Γ(34 )
Γ(14 )
√ 2π
) (14.1.4)
illustrates this point. Vardi [82] describes a good amount of mathematics involved in evaluating (14.1.4). The integral is first interpreted in terms of the derivative of the L-function
L(s) = 1− 1
+ 1
− 1
− · · · (14.1.5)
ln ln tanx dx = −πγ 4
+ L′(1). (14.1.6)
Here γ is Euler’s constant. Then L′(1) is computed in terms of the gamma function. This is an unexpected procedure.
