## ABSTRACT

CHAPTER 8

8.1. Introduction

The table of integrals [35] contains many entries of the form

(8.1.1)

R1(x) lnR2(x) dx

where R1 and R2 are rational functions. Some of these examples have appeared in previous papers: entry 4.291.1

(8.1.2)

ln(1 + x)

x dx =

pi2

as well as entry 4.291.2

(8.1.3)

ln(1− x) x

dx = −pi 2

have been established in [10], entry 4.212.7

(8.1.4)

lnx dx

(1 + lnx)2 = e

2 − 1

appears in [8] and entry 4.231.11

(8.1.5)

lnx dx

x2 + a2 = pi ln a

4a − G a ,

where

(8.1.6) G = ∞∑ k=0

(−1)k (2k + 1)2

is the Catalan constant, has appeared in [20]. The value of entry 4.233.1

(8.1.7)

ln x dx

x2 + x+ 1 =

[ 2pi2

3 − ψ′

( 1

)] ,

where ψ(x) = Γ′(x)/Γ(x) is the digamma function, was established in [54]. A standard trick employed in the evaluations of integrals over [0,∞) is to

transform the interval [1,∞) back to [0, 1] via t = 1/x. This gives

(8.1.8)

∫ ∞

R(x) lnx dx =

∫ 1

[ R(x)− 1

2 R

( 1

)] dx.