Dit artikel bevat een lijst van integralen van logaritmische functies. Het is met integralen mogelijk totalen te berekenen, zoals de totale oppervlakte onder een grafiek. De logaritme in de volgende integralen is steeds de natuurlijke logaritme. De reële logaritme is alleen gedefinieerd voor
. Er wordt van alle integralen de primitieve functie zonder integratieconstante gegeven.
![{\displaystyle \int \ln ax\ \mathrm {d} x=x\ln ax-x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db051c33581f74999d2290440d83c2986e3446c5)
![{\displaystyle \int \ln(ax+b)\ \mathrm {d} x={\frac {(ax+b)\ln(ax+b)-(ax)}{a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f31560552965c4d48d5b72ca0c82f27f78f47c59)
![{\displaystyle \int (\ln x)^{2}\ \mathrm {d} x=x(\ln x)^{2}-2x\ln x+2x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96ebbd2eea8b1bde0c6a2e91d08a9c37f8ee0a38)
![{\displaystyle \int (\ln x)^{n}\ \mathrm {d} x=x\sum _{k=0}^{n}(-1)^{n-k}{\frac {n!}{k!}}(\ln x)^{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a84a49d2b05a43c5205849b9c73fe58a12b56796)
![{\displaystyle \int {\frac {\mathrm {d} x}{\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/77dfe141f581f8230a332ed8a6ba0d67b04d77ed)
![{\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{voor }}n\neq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ee41cd744e312a64f06ec888ce3649f89f1d943)
![{\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{voor }}m\neq -1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46b8c40a2ea3ca611c199b906421dfc47c5470dc)
![{\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{voor }}m\neq -1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01f9dc7f976c697e81216cc07c34bd1779c83b96)
![{\displaystyle \int {\frac {(\ln x)^{n}\ \mathrm {d} x}{x}}={\frac {(\ln x)^{n+1}}{n+1}}\qquad {\mbox{voor }}n\neq -1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fc4bb2d39a9bcce70aae64e95e1a8b0658ba60a)
![{\displaystyle \int {\frac {\ln {x^{n}}\ \mathrm {d} x}{x}}={\frac {(\ln {x^{n}})^{2}}{2n}}\qquad {\mbox{voor }}n\neq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1fdcac7d9c277406ecbd5fa9d5c6d46280c3d27e)
![{\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{voor }}m\neq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/178c99895495aabd7e1ce62d0b87c0c1ea766737)
![{\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{voor }}m\neq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9a948dbf980142a164a21d10a1f8a9980c2751)
![{\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{voor }}n\neq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8380597d34a1658839a063a5a19cfac8b0b336ca)
![{\displaystyle \int {\frac {\mathrm {d} x}{x\ln x}}=\ln |\ln x|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84be13687403ee6e9f0e63a516e503e4e80f2696)
![{\displaystyle \int {\frac {\mathrm {d} x}{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/2b31106748050e62af4acfff4f6761685b50deeb)
![{\displaystyle \int {\frac {\mathrm {d} x}{x(\ln x)^{n}}}=-{\frac {1}{(n-1)(\ln x)^{n-1}}}\qquad {\mbox{voor }}n\neq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c171f06e66780b40e52b6cffbb53e6eacf7df7eb)
![{\displaystyle \int \ln(x^{2}+a^{2})\ \mathrm {d} x=x\ln(x^{2}+a^{2})-2x+2a\tan ^{-1}{\frac {x}{a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d41de0c9383ce90e69c7c9d70d0da25bc6c51e70)
![{\displaystyle \int {\frac {x}{x^{2}+a^{2}}}\ln(x^{2}+a^{2})\ \mathrm {d} x={\frac {1}{4}}\ln ^{2}(x^{2}+a^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23b40cd16678958032d08c6562496fc682209e07)
![{\displaystyle \int \ln \left({\sqrt {x^{n}}}\right)\ \mathrm {d} x={\frac {x}{2}}(\ln(x^{n})-n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a51118d715c23b69dc15710c7000bd7b7ae7216)
![{\displaystyle \int \sin(\ln x)\ \mathrm {d} x={\frac {x}{2}}(\sin(\ln x)-\cos(\ln x))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b88fd3763372a39a157e6cf90e13c4ad3462504)
![{\displaystyle \int \cos(\ln x)\ \mathrm {d} x={\frac {x}{2}}(\sin(\ln x)+\cos(\ln x))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4226964855300bc5623a22cf40092c2fa45cbc8)
![{\displaystyle \int e^{x}\left(x\ln x-x-{\frac {1}{x}}\right)\ \mathrm {d} x=e^{x}(x\ln x-x-\ln x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d86bd47aaa70b72f8dcab25abd9a721a95be4ac)
![{\displaystyle \int {\frac {1}{e^{x}}}\left({\frac {1}{x}}-\ln x\right)\ \mathrm {d} x={\frac {\ln x}{e^{x}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7469dd4c982e3ca6162c19c03927c271c20517c4)
![{\displaystyle \int (\ln x)^{x}\ \mathrm {d} x=(\ln x)^{x-1}+(\ln(\ln x))(\ln x)^{x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e08a4d3aef8a49701cba49aafd81b852e1a320d)
![{\displaystyle \int e^{x}\left({\frac {1}{\ln x}}-{\frac {1}{x\ln ^{2}x}}\right)\ \mathrm {d} x={\frac {e^{x}}{\ln x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a2dacb67bce57d431d31fcca175c34351336f2e)