ABSTRACT
CHAPTER 15
15.1. The list
This is the list of formulas in Gradshteyn and Ryzhik [35] that are established in this volume. The sections are named as in the table.
Section 2.14. Forms containing the binomial 1± xn
Subsection 2.148
2.148.3
∫ dx
(1 + x2)n =
2n− 2 x
(1 + x2)n−1 +
2n− 3 2n− 2
∫ dx
(1 + x2)n−1 28
Section 2.42 Powers of sinh x, cosh x, tanh x, and coth x
Subsection 2.423
2.423.9
∫ dx
coshx = 2 arctan(ex) 73
Section 3.13−3.17. Expressions that can be reduced to square roots of third-and fourth-degree polynomials and their products with
rational functions
Subsection 3.166
3.166.16
dx√ 1− x4 =
4 √ 2pi
{ Γ
( 1
)}2 2
3.166.18
x2 dx√ 1− x4 =
1√ 2pi
{ Γ
( 3
)}2 2
Section 3.19−3.23. Combinations of powers of x and powers of binomials of the form (α+ βx)
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“book2” — ✐
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Subsection 3.194
xµ−1 dx (1 + βx)ν
= uµ
µ 2F1 (ν, µ; 1 + µ;−βu) 3.194.1 63∫ ∞
xµ−1 dx (1 + βx)ν
= uµ−ν
βu(ν − µ)2F1
× ( ν, ν − µ; ν − µ+ 1;− 1
βu
) 3.194.2 68∫ ∞
xµ−1 dx (1 + βx)ν
= β−νB(µ, ν − µ) 3.194.3 172∫ u 0
xµ−1 dx 1 + βx
= uµ
µ 2F1(1, µ; 1 + µ;−βµ) 3.194.5 63∫ 1
xn−1 dx (1 + x)m
= 2−n ∞∑ k=0
( m− n− 1
k
) (−2)−k n+ k
3.194.8 140
Subsection 3.196
(x+ β)ν(u − x)µ−1 dx = β νuµ
µ 2F1
( 1,−ν; 1 + µ;−u
β
) 3.196.1 63
Subsection 3.197
x−λ(x+ β)ν(x− u)µ−1 dx
= (β + u)µ+ν
uλ B(λ− µ− ν, µ)2F1
× ( λ, µ;λ− µ;−β
u
) 3.197.2 69
xλ−1(1− x)µ−1(1− βx)−ν dx = B(λ, µ)2F1(ν, λ;λ+ µ;β) 3.197.3 62∫ 1
xµ−1(1− x)ν−1(1 + ax)−µ−ν dx
= (1 + a)−µB(µ, ν) 3.197.4 62∫ ∞ 0
xλ−1(1 + x)ν(1 + αx)µ dx
= B(λ,−µ− ν − λ)2F1 (−µ, λ;−µ− ν; 1− α) 3.197.5 68
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“book2” — ✐
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xλ−ν(x− 1)ν−µ−1(αx − 1)−λ dx
= B(µ, ν − µ)
αλ 2F1(ν, µ;λ;α
−1) 3.197.6 68∫ ∞ 0
xµ−1/2(x+ a)−µ(x+ b)−µ dx
= √ pi( √ a+
√ b)1−2µ
Γ(µ− 1/2) Γ(µ)
3.197.7 69∫ u 0
xν−1(x + α)λ(u− x−)µ−1 dx
= αλuµ+ν−1B(µ, ν)2F1 ( −λ, ν;µ+ ν;−u
α
) 3.197.8 68∫ ∞
xλ−1(1 + x)−µ+ν(x+ β)−ν dx
= B(µ− λ, λ) 2F1(ν, µ− λ;µ; 1− β) 3.197.9 68∫ 1 0
xq−1 dx (1− x)q(1 + px) =
pi
(1 + p)q cosec piq 3.197.10 62
Subsection 3.198
xµ−1(1− x)ν−1 dx [ax+ b(1− x) + x]µ+ν =
B(µ, ν)
(a+ c)µ (b+ c)ν 3.198 63
Subsection 3.199
(x − a)µ−1(b − x)ν−1 (x − c)µ+ν dx =
(b− a)µ+ν−1 (b− c)µ(a− c)νB(µ, ν) 3.199 64
Subsection 3.227
xν−1(β + x)1−µ
γ + x dx
= γν−1
βµ−1 B(ν, µ− ν)2F1
( µ− 1, ν;µ; 1− γ
β
) 3.227.1 68
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Subsection 3.231
xp−1 − x−p 1− x dx = pi cot pip 3.231.1 171∫ 1
xµ−1 − xν−1 1− x dx = ψ(ν) − ψ(µ) 3.231.5 78∫ ∞
xp−1 − xq−1 1− x dx = pi(cot pip− cot piq) 3.231.6 169
Section 3.24−3.27. Powers of x, of binomials of the form α+ βxp and of polynomials in x
Subsection 3.241
PV
xp−1 dx 1− xq =
pi
q cot
pip
q 3.241.3 172
Subsection 3.265
1− xµ−1 1− x = ψ(µ) +C 3.265 90
Subsection 3.271
( xp − x−p 1− x
)2 dx = 2(1− 2ppi cot 2ppi) 3.271.1 5
Section 3.3− 3.4. Exponential functions Subsection 3.31. Exponential functions
Subsection 3.310.
