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# An elementary proof of entry 3.411.5

DOI link for An elementary proof of entry 3.411.5

An elementary proof of entry 3.411.5 book

# An elementary proof of entry 3.411.5

DOI link for An elementary proof of entry 3.411.5

An elementary proof of entry 3.411.5 book

## 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.