Lista de integrais de funções logarítmicas

Segue uma lista de integrais de funções logarítmicas. Para uma lista geral de integrais, ver tábua de integrais.

Nota: assume-se x>0 neste artigo.

${\displaystyle \int \log _{a}x\,dx=x\log _{a}x-{\frac {x}{\log a}+C}$

${\displaystyle \int \ln cx\;dx=x\ln cx-x}$

${\displaystyle \int \ln(ax+b)\;dx={\frac {(ax+b)\ln(ax+b)-(ax+b)}{a}$

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

${\displaystyle \int (\ln x)^{n}\;dx=x\sum _{k=0}^{n}(-1)^{k}{\frac {n!(\ln x)^{n-k}{(n-k)!}$

${\displaystyle \int {\frac {dx}{\ln x}=\ln |\ln x|+\ln x+\sum _{k=2}^{\infty }{\frac {(\ln x)^{k}{k\cdot k!}$

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

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

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

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

${\displaystyle \int {\frac {\ln {x^{n}\;dx}{x}={\frac {(\ln {x^{n})^{2}{2n}\qquad {\mbox{(for }n\neq 0{\mbox{)}$

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

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

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

${\displaystyle \int {\frac {dx}{x\ln x}=\ln \left|\ln x\right|}$

${\displaystyle \int {\frac {dx}{x^{n}\ln x}=\ln \left|\ln x\right|+\sum _{k=1}^{\infty }(-1)^{k}{\frac {(n-1)^{k}(\ln x)^{k}{k\cdot k!}$

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

${\displaystyle \int \ln(x^{2}+a^{2})\;dx=x\ln(x^{2}+a^{2})-2x+2a\tan ^{-1}{\frac {x}{a}$

${\displaystyle \int {\frac {x}{x^{2}+a^{2}\ln(x^{2}+a^{2})\;dx={\frac {1}{4}\ln ^{2}(x^{2}+a^{2})}$

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

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

${\displaystyle \int e^{x}\left(x\ln x-x-{\frac {1}{x}\right)\;dx=e^{x}(x\ln x-x-\ln x)}$

${\displaystyle \int {\frac {1}{e^{x}\left({\frac {1}{x}-\ln x\right)\;dx={\frac {\ln x}{e^{x}$

${\displaystyle \int e^{x}\left({\frac {1}{\ln x}-{\frac {1}{x\ln ^{2}x}\right)\;dx={\frac {e^{x}{\ln x}$

• Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 1964. A few integrals are listed on page 69 in this classic book.