Anexo:Integrales de funciones logarítmicas

La siguiente es una lista de integrales de funciones logarítmicas.

Se asume en todos los casos que $x>0$.

La mayoría de ellas puede verificarse utilizando el método de integración por partes:

${\begin{aligned}&\int \ln x\,dx=x\ln x-x\\&\int (\ln x)^{2}\;dx=x(\ln x)^{2}-2x\ln x+2x\\&\int (\ln x)^{n}\;dx=x(\ln x)^{n}-n\int (\ln x)^{n-1}dx\quad {\mbox{para }}n\neq 1\\&\int {\frac {dx}{\ln x}}=\ln |\ln x|+\ln x+\sum _{i=2}^{\infty }{\frac {(\ln x)^{i}}{i\cdot i!}}\\&\int {\frac {dx}{(\ln x)^{n}}}=-{\frac {x}{(n-1)(\ln x)^{n-1}}}+{\frac {1}{n-1}}\int {\frac {dx}{(\ln x)^{n-1}}}\quad {\mbox{para }}n\neq 1\\&\int x^{m}\ln x\;dx=x^{m+1}\left({\frac {\ln x}{m+1}}-{\frac {1}{(m+1)^{2}}}\right)\quad {\mbox{para }}m\neq 1\\&\int x^{m}(\ln x)^{n}\;dx={\frac {x^{m+1}(\ln x)^{n}}{m+1}}-{\frac {n}{m+1}}\int x^{m}(\ln x)^{n-1}dx\quad {\mbox{para }}m,n\neq 1\\&\int {\frac {(\ln x)^{n}}{x}}\;dx={\frac {(\ln x)^{n+1}}{n+1}}\quad {\mbox{para }}n\neq 1\\&\int {\frac {\ln x}{x^{m}}}\;dx=-{\frac {\ln x}{(m-1)x^{m-1}}}-{\frac {1}{(m-1)^{2}x^{m-1}}}\quad {\mbox{para }}m\neq 1\\&\int {\frac {(\ln x)^{n}\;dx}{x^{m}}}=-{\frac {(\ln x)^{n}}{(m-1)x^{m-1}}}+{\frac {n}{m-1}}\int {\frac {(\ln x)^{n-1}dx}{x^{m}}}\quad {\mbox{para }}m,n\neq 1\\&\int {\frac {x^{m}\;dx}{(\ln x)^{n}}}=-{\frac {x^{m+1}}{(n-1)(\ln x)^{n-1}}}+{\frac {m+1}{n-1}}\int {\frac {x^{m}dx}{(\ln x)^{n-1}}}\quad {\mbox{para }}n\neq 1\\&\int {\frac {dx}{x\ln x}}=\ln |\ln x|\\&\int {\frac {dx}{x^{n}\ln x}}=\ln |\ln x|+\sum _{i=1}^{\infty }(-1)^{i}{\frac {(n-1)^{i}(\ln x)^{i}}{i\cdot i!}}\\&\int {\frac {dx}{x(\ln x)^{n}}}=-{\frac {1}{(n-1)(\ln x)^{n-1}}}\quad {\mbox{para }}n\neq 1\\&\int \operatorname {sen}(\ln x)\;dx={\frac {x}{2}}(\operatorname {sen}(\ln x)-\cos(\ln x))\\&\int \cos(\ln x)\;dx={\frac {x}{2}}(\operatorname {sen}(\ln x)+\cos(\ln x))\end{aligned}}$