§4.10 Integrals

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Referenced by:: §1.4(iv)
Permalink:: http://dlmf.nist.gov/4.10
See also:: Annotations for Ch.4

§4.10(i) Logarithms

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Keywords:: integrals, logarithm function
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).
Permalink:: http://dlmf.nist.gov/4.10.i
See also:: Annotations for §4.10 and Ch.4

4.10.1		$\int\frac{\,\mathrm{d}z}{z}=\ln z,$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : 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 Encodings: TeX, pMML, png See also: Annotations for §4.10(i), §4.10 and Ch.4

4.10.2		$\int\ln z\,\mathrm{d}z=z\ln z-z,$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable A&S Ref: 4.1.49 Permalink: http://dlmf.nist.gov/4.10.E2 Encodings: TeX, pMML, png See also: Annotations for §4.10(i), §4.10 and Ch.4

4.10.3		$\int z^{n}\ln z\,\mathrm{d}z=\frac{z^{n+1}}{n+1}\ln z-\frac{z^{n+1}}{(n+1)^{2}},$
$n\neq-1$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : 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 Encodings: TeX, pMML, png See also: Annotations for §4.10(i), §4.10 and Ch.4

4.10.4		$\int\frac{\,\mathrm{d}z}{z\ln z}=\ln\left(\ln z\right),$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : 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 Encodings: TeX, pMML, png See also: Annotations for §4.10(i), §4.10 and Ch.4

4.10.5		$\int_{0}^{1}\frac{\ln t}{1-t}\,\mathrm{d}t=-\frac{\pi^{2}}{6},$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\ln\NVar{z}$ : principal branch of logarithm function A&S Ref: 4.1.55 Referenced by: §4.10(i) Permalink: http://dlmf.nist.gov/4.10.E5 Encodings: TeX, pMML, png See also: Annotations for §4.10(i), §4.10 and Ch.4

4.10.6		$\int_{0}^{1}\frac{\ln t}{1+t}\,\mathrm{d}t=-\frac{\pi^{2}}{12},$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\ln\NVar{z}$ : principal branch of logarithm function A&S Ref: 4.1.56 Referenced by: §4.10(i) Permalink: http://dlmf.nist.gov/4.10.E6 Encodings: TeX, pMML, png See also: Annotations for §4.10(i), §4.10 and Ch.4

4.10.7		$\pvint_{0}^{x}\frac{\,\mathrm{d}t}{\ln t}=\operatorname{li}\left(x\right),$
$x>1$ .
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{li}\left(\NVar{x}\right)$ : logarithmic integral, $\ln\NVar{z}$ : principal branch of logarithm function, $\pvint_{\NVar{a}}^{\NVar{b}}$ : 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 Encodings: TeX, pMML, png See also: Annotations for §4.10(i), §4.10 and Ch.4

The left-hand side of (4.10.7) is a Cauchy principal value (§1.4(v)). For $\operatorname{li}\left(x\right)$ see §6.2(i).

§4.10(ii) Exponentials

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Keywords:: exponential function, integrals
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).
Permalink:: http://dlmf.nist.gov/4.10.ii
See also:: Annotations for §4.10 and Ch.4

For $a,b\neq 0$ ,

4.10.8		$\int e^{az}\,\mathrm{d}z=\frac{e^{az}}{a},$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : 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 Encodings: TeX, pMML, png See also: Annotations for §4.10(ii), §4.10 and Ch.4

4.10.9		$\int\frac{\,\mathrm{d}z}{e^{az}+b}=\frac{1}{ab}(az-\ln\left(e^{az}+b\right)),$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\ln\NVar{z}$ : 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 Encodings: TeX, pMML, png See also: Annotations for §4.10(ii), §4.10 and Ch.4

4.10.10		$\int\frac{e^{az}-1}{e^{az}+1}\,\mathrm{d}z=\frac{2}{a}\ln\left(e^{az/2}+e^{-az% /2}\right),$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\ln\NVar{z}$ : 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 Encodings: TeX, pMML, png See also: Annotations for §4.10(ii), §4.10 and Ch.4

4.10.11		$\int_{-\infty}^{\infty}e^{-cx^{2}}\,\mathrm{d}x=\sqrt{\frac{\pi}{c}},$
$\Re c>0$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : 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 Encodings: TeX, pMML, png See also: Annotations for §4.10(ii), §4.10 and Ch.4

4.10.12		$\int_{0}^{\ln 2}\frac{xe^{x}}{e^{x}-1}\,\mathrm{d}x=\frac{\pi^{2}}{12},$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function and $x$ : real variable Referenced by: §4.10(ii) Permalink: http://dlmf.nist.gov/4.10.E12 Encodings: TeX, pMML, png See also: Annotations for §4.10(ii), §4.10 and Ch.4

4.10.13		$\int_{0}^{\infty}\frac{\,\mathrm{d}x}{e^{x}+1}=\ln 2.$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\ln\NVar{z}$ : 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 See also: Annotations for §4.10(ii), §4.10 and Ch.4

§4.10(iii) Compendia

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Permalink:: http://dlmf.nist.gov/4.10.iii
See also:: Annotations for §4.10 and Ch.4

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 (2015, 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