in Wikipedia, die vrye ensiklopedie
Hier volg 'n lys van integrale (anti-afgeleide funksies) van logaritmiese funksies. Vir 'n volledige lys van integrale funksies, sien lys van integrale.
Nota: x > 0 word in hierdie artikel deurgaans veronderstel, en die konstante van integrasie word vir eenvoudigheid weggelaat.
![{\displaystyle \int \log _{a}x\,dx={\frac {x\ln x-x}{\ln a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8c34a7baf2dc0faf8363e49702212bd01dd3bec)
![{\displaystyle \int \ln(ax)\,dx=x\ln(ax)-x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81ee14f755cee0a446b470255343e60f3c80d982)
![{\displaystyle \int \ln(ax+b)\,dx={\frac {(ax+b)\ln(ax+b)-ax}{a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7163e09af661a4c088c0dde0330c586771c84665)
![{\displaystyle \int (\ln x)^{2}\,dx=x(\ln x)^{2}-2x\ln x+2x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5853a2d856005e03dbeb45c5e1316ad96987fec)
![{\displaystyle \int (\ln x)^{n}\,dx=x\sum _{k=0}^{n}(-1)^{n-k}{\frac {n!}{k!}}(\ln x)^{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46f4ac1cb2650cedf436b2bf6ac167e46781461d)
![{\displaystyle \int {\frac {dx}{\ln x}}=\ln |\ln x|+\ln x+\sum _{k=2}^{\infty }{\frac {(\ln x)^{k}}{k\cdot k!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b80db5839b7b4074c717480feeac2550240c486)
, die logaritmiese integraal.
![{\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{(vir }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8c379b87974b2a6b1aa2a9f8899af01c93fb6a6)
![{\displaystyle \int x^{m}\ln x\,dx=x^{m+1}\left({\frac {\ln x}{m+1}}-{\frac {1}{(m+1)^{2}}}\right)\qquad {\mbox{(vir }}m\neq -1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a85a34555dc4c713c5a17bcb8fbc536902741960)
![{\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{(vir }}m\neq -1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8df581eb13297941041b718aa4c8362b17e49a79)
![{\displaystyle \int {\frac {(\ln x)^{n}\,dx}{x}}={\frac {(\ln x)^{n+1}}{n+1}}\qquad {\mbox{(vir }}n\neq -1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ccd386ed9438eb56200a7fb4ca81001818dbd415)
![{\displaystyle \int {\frac {\ln {x^{n}}\,dx}{x}}={\frac {(\ln {x^{n}})^{2}}{2n}}\qquad {\mbox{(vir }}n\neq 0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99577b6720aeac0ecd327b6ef5558f9ff5fe420a)
![{\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{(vir }}m\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c51ccee58a550c1403d8904bcb13026e3269f9a4)
![{\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{(vir }}m\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b12775389b07c4e2254f28a3b8129d2056ec2d2d)
![{\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{(vir }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e8ebcae7b1891539752794e9f5145c3b74b52a1)
![{\displaystyle \int {\frac {dx}{x\ln x}}=\ln \left|\ln x\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fce91dfa19693fbe5883d150963bed432e06d047)
, etc.
![{\displaystyle \int {\frac {dx}{x\ln \ln x}}=\operatorname {li} (\ln x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/180e391e734340059024d58f0396fd1c69fca77d)
![{\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!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/500ec1ec23b97b79baf39b8febbb589f11e37b82)
![{\displaystyle \int {\frac {dx}{x(\ln x)^{n}}}=-{\frac {1}{(n-1)(\ln x)^{n-1}}}\qquad {\mbox{(vir }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9eacbbb4f08977cc34ab07bfbd4b879e4396bcf)
![{\displaystyle \int \ln(x^{2}+a^{2})\,dx=x\ln(x^{2}+a^{2})-2x+2a\tan ^{-1}{\frac {x}{a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb444c640d10b7d12aec5cabd50c55c0a957b820)
![{\displaystyle \int {\frac {x}{x^{2}+a^{2}}}\ln(x^{2}+a^{2})\,dx={\frac {1}{4}}\ln ^{2}(x^{2}+a^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8473d4e4b11e413ddb0d6965f0d0adfe7a58015e)
Integrale wat logaritmiese en trigonometriese funksies bevat
[wysig | wysig bron]
![{\displaystyle \int \sin(\ln x)\,dx={\frac {x}{2}}(\sin(\ln x)-\cos(\ln x))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33c6c667b8cfe2a07263f42d1392cc2dfdd65cc5)
![{\displaystyle \int \cos(\ln x)\,dx={\frac {x}{2}}(\sin(\ln x)+\cos(\ln x))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/798da7bc36fe2c1edd580c90342c1cde8086df4d)
Integrale wat logaritmiese en eksponensiële funksies bevat
[wysig | wysig bron]
![{\displaystyle \int e^{x}\left(x\ln x-x-{\frac {1}{x}}\right)\,dx=e^{x}(x\ln x-x-\ln x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8632f3aea3bdd5b6c25fa45547d2d3359960dfbc)
![{\displaystyle \int {\frac {1}{e^{x}}}\left({\frac {1}{x}}-\ln x\right)\,dx={\frac {\ln x}{e^{x}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/617a944320af8e067a09c96720fa8ef436f4a69f)
![{\displaystyle \int e^{x}\left({\frac {1}{\ln x}}-{\frac {1}{x(\ln x)^{2}}}\right)\,dx={\frac {e^{x}}{\ln x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/021b378a21d01e5131cc4fa11466d9ada8990f58)
Vir
opeenvolgende integrasies, veralgemeen die formule
![{\displaystyle \int \ln x\,dx=x(\ln x-1)+C_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45ced9abfbe033de1dab09c1dfa704755885c317)
na
![{\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!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c34d7df20d2d235e0165f209c08efc563371bb3)