## 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)

“book2” — ✐

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

“book2” — ✐

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

“book2” — ✐

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.