Список на интеграли на логаритамски функции

**Списoк на [[интеграл]]и на логаритамски [[функција|функции]]**:

*x*&gt;0 важи за сите интеграли во оваа статија.

: \int\ln cx\,dx = x\ln cx - x

: \int (\ln x)^2\; dx = x(\ln x)^2 - 2x\ln x + 2x

: \int (\ln cx)^n\; dx = x(\ln cx)^n - n\int (\ln cx)^{n-1} dx \qquad\mbox{(for }n\neq 1\mbox{)}

: \int \frac{dx}{\ln x} = \ln|\ln x| + \ln x + \sum^\infty_{i=2}\frac{(\ln x)^i}{i\cdot i!}

: \int \frac{dx}{(\ln x)^n} = -\frac{x}{(n-1)(\ln x)^{n-1}} + \frac{1}{n-1}\int\frac{dx}{(\ln x)^{n-1}} \qquad\mbox{(for }n\neq 1\mbox{)}

: \int x^m\ln x\;dx = x^{m+1}\left(\frac{\ln x}{m+1}-\frac{1}{(m+1)^2}\right) \qquad\mbox{(for }m\neq 1\mbox{)}

: \int x^m (\ln x)^n\; dx = \frac{x^{m+1}(\ln x)^n}{m+1} - \frac{n}{m+1}\int x^m (\ln x)^{n-1} dx  \qquad\mbox{(for }m,n\neq 1\mbox{)}

: \int \frac{(\ln x)^n\; dx}{x} = \frac{(\ln x)^{n+1}}{n+1}  \qquad\mbox{(for }n\neq 1\mbox{)}

: \int \frac{\ln x\,dx}{x^m} = -\frac{\ln x}{(m-1)x^{m-1}}-\frac{1}{(m-1)^2 x^{m-1}} \qquad\mbox{(for }m\neq 1\mbox{)}

: \int \frac{(\ln x)^n\; dx}{x^m} = -\frac{(\ln x)^n}{(m-1)x^{m-1}} + \frac{n}{m-1}\int\frac{(\ln x)^{n-1} dx}{x^m} \qquad\mbox{(for }m,n\neq 1\mbox{)}

: \int \frac{x^m\; dx}{(\ln x)^n} = -\frac{x^{m+1}}{(n-1)(\ln x)^{n-1}} + \frac{m+1}{n-1}\int\frac{x^m dx}{(\ln x)^{n-1}}  \qquad\mbox{(for }n\neq 1\mbox{)}

: \int \frac{dx}{x\ln x} = \ln|\ln x|

: \int \frac{dx}{x^n\ln x} = \ln|\ln x| + \sum^\infty_{i=1} (-1)^i\frac{(n-1)^i(\ln x)^i}{i\cdot i!}

: \int \frac{dx}{x (\ln x)^n} = -\frac{1}{(n-1)(\ln x)^{n-1}} \qquad\mbox{(for }n\neq 1\mbox{)}

: \int \sin (\ln x)\;dx = \frac{x}{2}(\sin (\ln x) - \cos (\ln x))

: \int \cos (\ln x)\;dx = \frac{x}{2}(\sin (\ln x) + \cos (\ln x))

## Литература

* Milton Abramowitz and Irene Stegun, editors. *Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*.
* I.S. Gradshteyn (И. С. Градштейн), I.M. Ryzhik (И. М. Рыжик); Alan Jeffrey, Daniel Zwillinger, editors. *Table of Integrals, Series, and Products*, seventh edition. Academic Press, 2007. ISBN 978-0-12-373637-6. Errata. *(Several previous editions as well.)*
* A.P. Prudnikov (А. П. Прудников), Yu.A. Brychkov (Ю. А. Брычков), O.I. Marichev (О. И. Маричев). *Integrals and Series*. First edition (Russian), volume 1–5, Nauka, 1981−1986. First edition (English, translated from the Russian by N.M. Queen), volume 1–5, Gordon & Breach Science Publishers/CRC Press, 1988–1992, ISBN 2-88124-097-6. Second revised edition (Russian), volume 1–3, Fiziko-Matematicheskaya Literatura, 2003.
* Yu.A. Brychkov (Ю. А. Брычков), *Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas*. Russian edition, Fiziko-Matematicheskaya Literatura, 2006. English edition, Chapman & Hall/CRC Press, 2008, ISBN 1-58488-956-X.
* Daniel Zwillinger. *CRC Standard Mathematical Tables and Formulae*, 31st edition. Chapman & Hall/CRC Press, 2002. ISBN 1-58488-291-3. *(Many earlier editions as well.)*
* Meyer Hirsch, Integral Tables, Or, A Collection of Integral Formulae (Baynes and son, London, 1823) [English translation of *Integraltafeln*] 
* Benjamin O. Pierce A short table of integrals - revised edition (Ginn & co., Boston, 1899)

Категорија:Математичка анализа
Категорија:Интегрално сметање
Категорија:Интеграли