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.
