§4.10 Integrals

Referenced by:: ¶ ‣ §1.4(iv)
Permalink:: http://dlmf.nist.gov/4.10

§4.10(i) Logarithms
§4.10(ii) Exponentials
§4.10(iii) Compendia

§4.10(i) Logarithms

Notes:: (4.10.1)–(4.10.4) can be verified by differentiation. For (4.10.5) and (4.10.6), expand by the geometric series and integrate term by term to get a series which can be summed by Andrews et al. (1999, p. 12).
Keywords:: integrals, logarithm function
Permalink:: http://dlmf.nist.gov/4.10.i

4.10.1 $\int\frac{dz}{z}=\mathop{\ln\/}\nolimits z,$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $z$ : complex variable
A&S Ref:: 4.1.48
Referenced by:: §4.10(i)
Permalink:: http://dlmf.nist.gov/4.10.E1
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4.10.2 $\int\mathop{\ln\/}\nolimits zdz=z\mathop{\ln\/}\nolimits z-z,$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $z$ : complex variable
A&S Ref:: 4.1.49
Permalink:: http://dlmf.nist.gov/4.10.E2
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4.10.3 $\int z^{n}\mathop{\ln\/}\nolimits zdz=\frac{z^{{n+1}}}{n+1}\mathop{\ln\/}\nolimits z-\frac{z^{{n+1}}}{(n+1)^{2}},$ $n\neq-1$ ,

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $n$ : integer and $z$ : complex variable
A&S Ref:: 4.1.50
Permalink:: http://dlmf.nist.gov/4.10.E3
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4.10.4 $\int\frac{dz}{z\mathop{\ln\/}\nolimits z}=\mathop{\ln\/}\nolimits\!\left(\mathop{\ln\/}\nolimits z\right),$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $z$ : complex variable
A&S Ref:: 4.1.52
Referenced by:: §4.10(i)
Permalink:: http://dlmf.nist.gov/4.10.E4
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4.10.5 $\int _{0}^{1}\frac{\mathop{\ln\/}\nolimits t}{1-t}dt=-\frac{\pi^{2}}{6},$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral and $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function
A&S Ref:: 4.1.55
Referenced by:: §4.10(i)
Permalink:: http://dlmf.nist.gov/4.10.E5
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4.10.6 $\int _{0}^{1}\frac{\mathop{\ln\/}\nolimits t}{1+t}dt=-\frac{\pi^{2}}{12},$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral and $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function
A&S Ref:: 4.1.56
Referenced by:: §4.10(i)
Permalink:: http://dlmf.nist.gov/4.10.E6
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4.10.7 $\pvint _{0}^{x}\frac{dt}{\mathop{\ln\/}\nolimits t}=\mathop{\mathrm{li}\/}\nolimits\!\left(x\right),$ $x>1$ .

Symbols:: $dx$ : differential of $x$ , $\mathop{\mathrm{li}\/}\nolimits\!\left(x\right)$ : logarithmic integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\pvint _{a}^{b}$ : Cauchy principal value and $x$ : real variable
A&S Ref:: 4.1.57
Referenced by:: §4.10(i)
Permalink:: http://dlmf.nist.gov/4.10.E7
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The left-hand side of (4.10.7) is a Cauchy principal value (§1.4(v)). For $\mathop{\mathrm{li}\/}\nolimits\!\left(x\right)$ see §6.2(i).

§4.10(ii) Exponentials

Notes:: (4.10.8)–(4.10.10) can be verified by differentiation. For (4.10.11) apply (5.4.6) and (5.9.1). To evaluate (4.10.12) and (4.10.13), expand by the geometric series and integrate term by term. The dilogarithm series which appears from (4.10.12) can be summed by Andrews et al. (1999, p. 105).
Keywords:: exponential function, integrals
Permalink:: http://dlmf.nist.gov/4.10.ii

For $a,b\neq 0$ ,

4.10.8 $\int e^{{az}}dz=\frac{e^{{az}}}{a},$

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $z$ : complex variable and $a\neq 0$ : complex
A&S Ref:: 4.2.54
Referenced by:: §4.10(ii)
Permalink:: http://dlmf.nist.gov/4.10.E8
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4.10.9 $\int\frac{dz}{e^{{az}}+b}=\frac{1}{ab}(az-\mathop{\ln\/}\nolimits\!\left(e^{{az}}+b\right)),$

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $z$ : complex variable, $a\neq 0$ : complex and $b\neq 0$ : complex
Permalink:: http://dlmf.nist.gov/4.10.E9
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4.10.10 $\int\frac{e^{{az}}-1}{e^{{az}}+1}dz=\frac{2}{a}\mathop{\ln\/}\nolimits\!\left(e^{{az/2}}+e^{{-az/2}}\right),$

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $z$ : complex variable and $a\neq 0$ : complex
Referenced by:: §4.10(ii)
Permalink:: http://dlmf.nist.gov/4.10.E10
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4.10.11 $\int _{{-\infty}}^{\infty}e^{{-cx^{2}}}dx=\sqrt{\frac{\pi}{c}},$ $\realpart{c}>0$ ,

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : real part, $c$ : real or complex constant and $x$ : real variable
Referenced by:: §36.2(ii), §4.10(ii)
Permalink:: http://dlmf.nist.gov/4.10.E11
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4.10.12 $\int _{0}^{{\mathop{\ln\/}\nolimits 2}}\frac{xe^{x}}{e^{x}-1}dx=\frac{\pi^{2}}{12},$

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $x$ : real variable
Referenced by:: §4.10(ii)
Permalink:: http://dlmf.nist.gov/4.10.E12
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4.10.13 $\int _{0}^{\infty}\frac{dx}{e^{x}+1}=\mathop{\ln\/}\nolimits 2.$

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $x$ : real variable
Referenced by:: §4.10(ii)
Permalink:: http://dlmf.nist.gov/4.10.E13
Encodings:: TeX, pMML, png

§4.10(iii) Compendia

Permalink:: http://dlmf.nist.gov/4.10.iii

Extensive compendia of indefinite and definite integrals of logarithms and exponentials include Apelblat (1983, pp. 16–47), Bierens de Haan (1939), Gröbner and Hofreiter (1949, pp. 107–116), Gröbner and Hofreiter (1950, pp. 52–90), Gradshteyn and Ryzhik (2000, Chapters 2–4), and Prudnikov et al. (1986a, §§1.3, 1.6, 2.3, 2.6).

§4.10 Integrals

Contents

§4.10(i) Logarithms

§4.10(ii) Exponentials

§4.10(iii) Compendia