La siguiente es una lista de integrales ( funciones antiderivadas ) de funciones logarítmicas . Para obtener una lista completa de funciones integrales, consulte lista de integrales .
Nota: En este artículo se supone x > 0 y la constante de integración se omite por simplicidad.
Integrales que involucran solo funciones logarítmicas
![{\displaystyle \int \log _{a}x\,dx=x\log _{a}x-{\frac {x}{\ln a}}={\frac {x}{\ln a}} (\lnx-1)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int \ln(ax)\,dx=x\ln(ax)-x=x(\ln(ax)-1)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int \ln(ax+b)\,dx={\frac {ax+b}{a}}(\ln(ax+b)-1)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int (\ln x)^{2}\,dx=x(\ln x)^{2}-2x\ln x+2x}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int (\ln x)^{n}\,dx=x\sum _{k=0}^{n}(-1)^{nk}{\frac {n!}{k!} }(\lnx)^{k}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {dx}{\ln x}}=\ln |\ln x|+\ln x+\sum _{k=2}^{\infty }{\frac {(\ln x )^{k}}{k\cdot k!}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
, la integral logarítmica .
![{\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{(para }}n\neq 1{ \mbox{)}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int \ln f(x)\,dx=x\ln f(x)-\int x{\frac {f'(x)}{f(x)}}\,dx\qquad {\ mbox{(para diferenciable }}f(x)>0{\mbox{)}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Integrales que involucran funciones logarítmicas y de potencia.
![{\displaystyle \int x^{m}\ln x\,dx=x^{m+1}\left({\frac {\ln x}{m+1}}-{\frac {1}{( m+1)^{2}}}\right)\qquad {\mbox{(para }}m\neq -1{\mbox{)}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\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{(para }}m\neq -1{\mbox {)}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {(\ln x)^{n}\,dx}{x}}={\frac {(\ln x)^{n+1}}{n+1}}\ qquad {\mbox{(para }}n\neq -1{\mbox{)}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\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{(para }}m\neq 1{\mbox{)}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\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{(para }}m\neq 1{\mbox{)}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\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{(para }}n\neq 1{\mbox{)}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {dx}{x\ln x}}=\ln \left|\ln x\right|}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
, etc.
![{\displaystyle \int {\frac {dx}{x\ln \ln x}}=\operatorname {li} (\ln x)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\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!}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {dx}{x(\ln x)^{n}}}=-{\frac {1}{(n-1)(\ln x)^{n-1}} }\qquad {\mbox{(para }}n\neq 1{\mbox{)}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int \ln(x^{2}+a^{2})\,dx=x\ln(x^{2}+a^{2})-2x+2a\tan ^{-1 }{\frac {x}{a}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {x}{x^{2}+a^{2}}}\ln(x^{2}+a^{2})\,dx={\frac {1} {4}}\ln ^{2}(x^{2}+a^{2})}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Integrales que involucran funciones logarítmicas y trigonométricas.
![{\displaystyle \int \sin(\ln x)\,dx={\frac {x}{2}}(\sin(\ln x)-\cos(\ln x))}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int \cos(\ln x)\,dx={\frac {x}{2}}(\sin(\ln x)+\cos(\ln x))}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Integrales que involucran funciones logarítmicas y exponenciales.
![{\displaystyle \int e^{x}\left(x\ln xx-{\frac {1}{x}}\right)\,dx=e^{x}(x\ln xx-\ln x) }](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {1}{e^{x}}}\left({\frac {1}{x}}-\ln x\right)\,dx={\frac {\ln x }{e^{x}}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int e^{x}\left({\frac {1}{\ln x}}-{\frac {1}{x(\ln x)^{2}}}\right)\, dx={\frac {e^{x}}{\ln x}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
n integraciones consecutivas
Para integraciones consecutivas, la fórmula![{\displaystyle n}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int \ln x\,dx=x(\ln x-1)+C_{0}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
generaliza a
![{\displaystyle \int \dotsi \int \ln x\,dx\dotsm dx={\frac {x^{n}}{n!}}\left(\ln \,x-\sum _ {k=1 }^{n}{\frac {1}{k}}\right)+\sum _{k=0}^{n-1}C_{k}{\frac {x^{k}}{k! }}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Ver también
Referencias