# §4.10 Integrals

## §4.10(i) Logarithms

 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 $\pvint_{0}^{x}\frac{\,\mathrm{d}t}{\ln t}=\operatorname{li}\left(x\right),$ $x>1$.

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

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 $\int\frac{\,\mathrm{d}z}{e^{az}+b}=\frac{1}{ab}(az-\ln\left(e^{az}+b\right)),$
 4.10.10 $\int\frac{e^{az}-1}{e^{az}+1}\,\mathrm{d}z=\frac{2}{a}\ln\left(e^{az/2}+e^{-az% /2}\right),$
 4.10.11 $\int_{-\infty}^{\infty}e^{-cx^{2}}\,\mathrm{d}x=\sqrt{\frac{\pi}{c}},$ $\Re c>0$,
 4.10.12 $\int_{0}^{\ln 2}\frac{xe^{x}}{e^{x}-1}\,\mathrm{d}x=\frac{\pi^{2}}{12},$
 4.10.13 $\int_{0}^{\infty}\frac{\,\mathrm{d}x}{e^{x}+1}=\ln 2.$

## §4.10(iii) Compendia

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).