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4 Elementary FunctionsLogarithm, Exponential, Powers

§4.10 Integrals

Contents

§4.10(i) Logarithms

4.10.1 dzz=lnz,
4.10.2 lnzdz=zlnz-z,
4.10.3 znlnzdz=zn+1n+1lnz-zn+1(n+1)2,
n-1,
4.10.4 dzzlnz=ln(lnz),
4.10.5 01lnt1-tdt=-π26,
4.10.6 01lnt1+tdt=-π212,
4.10.7 0xdtlnt=li(x),
x>1.

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

§4.10(ii) Exponentials

For a,b0,

4.10.8 eazdz=eaza,
4.10.9 dzeaz+b=1ab(az-ln(eaz+b)),
4.10.10 eaz-1eaz+1dz=2aln(eaz/2+e-az/2),
4.10.11 -e-cx2dx=πc,
c>0,
4.10.12 0ln2xexex-1dx=π212,
4.10.13 0dxex+1=ln2.

§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).