Seznam integrálů logaritmických funkcí

Toto je seznam integrálů (primitivních funkcí) logaritmických funkcí.

Poznámka: V následujících integrálech se předpokládá, že x>0.

${\displaystyle \int \ln cx\,\,\mathrm {d} x=x\ln(cx)-x}$

${\displaystyle \int (\ln x)^{2}\;\,\mathrm {d} x=x(\ln x)^{2}-2x\ln x+2x}$

${\displaystyle \int (\ln cx)^{n}\;\,\mathrm {d} x=x(\ln cx)^{n}-n\int (\ln cx)^{n-1}\,\mathrm {d} x}$

${\displaystyle \int {\frac {\,\mathrm {d} x}{\ln x}=\ln |\ln x|+\ln x+\sum _{i=2}^{\infty }{\frac {(\ln x)^{i}{i\cdot i!}$

${\displaystyle \int {\frac {\,\mathrm {d} x}{(\ln x)^{n}=-{\frac {x}{(n-1)(\ln x)^{n-1}+{\frac {1}{n-1}\int {\frac {\,\mathrm {d} x}{(\ln x)^{n-1}\qquad {\mbox{(pro }n\neq 1{\mbox{)}$

${\displaystyle \int x^{m}\ln x\;\,\mathrm {d} x=x^{m+1}\left({\frac {\ln x}{m+1}-{\frac {1}{(m+1)^{2}\right)\qquad {\mbox{(pro }m\neq -1{\mbox{)}$

${\displaystyle \int x^{m}(\ln x)^{n}\;\,\mathrm {d} x={\frac {x^{m+1}(\ln x)^{n}{m+1}-{\frac {n}{m+1}\int x^{m}(\ln x)^{n-1}\,\mathrm {d} x\qquad {\mbox{(pro }m\neq -1{\mbox{)}$

${\displaystyle \int {\frac {(\ln x)^{n}\;\,\mathrm {d} x}{x}={\frac {(\ln x)^{n+1}{n+1}\qquad {\mbox{(pro }n\neq -1{\mbox{)}$

${\displaystyle \int {\frac {\ln x\,\,\mathrm {d} x}{x^{m}=-{\frac {\ln x}{(m-1)x^{m-1}-{\frac {1}{(m-1)^{2}x^{m-1}\qquad {\mbox{(pro }m\neq 1{\mbox{)}$

${\displaystyle \int {\frac {(\ln x)^{n}\;\,\mathrm {d} x}{x^{m}=-{\frac {(\ln x)^{n}{(m-1)x^{m-1}+{\frac {n}{m-1}\int {\frac {(\ln x)^{n-1}\,\mathrm {d} x}{x^{m}\qquad {\mbox{(pro }m\neq 1{\mbox{)}$

${\displaystyle \int {\frac {x^{m}\;\,\mathrm {d} x}{(\ln x)^{n}=-{\frac {x^{m+1}{(n-1)(\ln x)^{n-1}+{\frac {m+1}{n-1}\int {\frac {x^{m}\,\mathrm {d} x}{(\ln x)^{n-1}\qquad {\mbox{(pro }n\neq 1{\mbox{)}$

${\displaystyle \int {\frac {\,\mathrm {d} x}{x\ln x}=\ln |\ln x|}$

${\displaystyle \int {\frac {\,\mathrm {d} x}{x^{n}\ln x}=\ln |\ln x|+\sum _{i=1}^{\infty }(-1)^{i}{\frac {(n-1)^{i}(\ln x)^{i}{i\cdot i!}$

${\displaystyle \int {\frac {\,\mathrm {d} x}{x(\ln x)^{n}=-{\frac {1}{(n-1)(\ln x)^{n-1}\qquad {\mbox{(pro }n\neq 1{\mbox{)}$

${\displaystyle \int \sin(\ln x)\;\,\mathrm {d} x={\frac {x}{2}(\sin(\ln x)-\cos(\ln x))}$

${\displaystyle \int \cos(\ln x)\;\,\mathrm {d} x={\frac {x}{2}(\sin(\ln x)+\cos(\ln x))}$

${\displaystyle \int e^{x}(x\ln x-x-{\frac {1}{x})\;\,\mathrm {d} x=e^{x}(x\ln x-x-\ln x)}$