Lijst van integralen van logaritmische functies

Dit artikel bevat een lijst van integralen van logaritmische functies. Nota bene dat x niet gedefinieerd is wanneer het kleiner of gelijk is aan nul. Tevens is de integratieconstante C weggelaten.

${\displaystyle \int \ln ax\,\mathrm {d} x=x\ln ax-x}$

${\displaystyle \int \ln(ax+b)\,\mathrm {d} x={\frac {(ax+b)\ln(ax+b)-(ax)}{a}}}$

${\displaystyle \int (\ln x)^{2}\,\mathrm {d} x=x(\ln x)^{2}-2x\ln x+2x}$

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

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

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

${\displaystyle \int x^{m}\ln x\,\mathrm {d} x=x^{m+1}\left({\frac {\ln x}{m+1}}-{\frac {1}{(m+1)^{2}}}\right)\qquad ({\mbox{voor }}m\neq -1)}$

${\displaystyle \int x^{m}(\ln x)^{n}\,\mathrm {d} x={\frac {x^{m+1}(\ln x)^{n}}{m+1}}-{\frac {n}{m+1}}\int x^{m}(\ln x)^{n-1}\,\mathrm {d} x\qquad ({\mbox{voor }}m\neq -1)}$

${\displaystyle \int {\frac {(\ln x)^{n}\,\mathrm {d} x}{x}}={\frac {(\ln x)^{n+1}}{n+1}}\qquad ({\mbox{voor }}n\neq -1)}$

${\displaystyle \int {\frac {\ln {x^{n}}\,\mathrm {d} x}{x}}={\frac {(\ln {x^{n}})^{2}}{2n}}\qquad ({\mbox{voor }}n\neq 0)}$

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

${\displaystyle \int {\frac {(\ln x)^{n}\,\mathrm {d} x}{x^{m}}}=-{\frac {(\ln x)^{n}}{(m-1)x^{m-1}}}+{\frac {n}{m-1}}\int {\frac {(\ln x)^{n-1}\,\mathrm {d} x}{x^{m}}}\qquad ({\mbox{voor }}m\neq 1)}$

${\displaystyle \int {\frac {x^{m}\,\mathrm {d} x}{(\ln x)^{n}}}=-{\frac {x^{m+1}}{(n-1)(\ln x)^{n-1}}}+{\frac {m+1}{n-1}}\int {\frac {x^{m}\,\mathrm {d} x}{(\ln x)^{n-1}}}\qquad ({\mbox{voor }}n\neq 1)}$

${\displaystyle \int {\frac {\mathrm {d} x}{x\ln x}}=\ln |\ln x|}$

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

${\displaystyle \int {\frac {\mathrm {d} x}{x(\ln x)^{n}}}=-{\frac {1}{(n-1)(\ln x)^{n-1}}}\qquad ({\mbox{voor }}n\neq 1)}$

${\displaystyle \int \ln(x^{2}+a^{2})\,\mathrm {d} x=x\ln(x^{2}+a^{2})-2x+2a\tan ^{-1}{\frac {x}{a}}}$

${\displaystyle \int {\frac {x}{x^{2}+a^{2}}}\ln(x^{2}+a^{2})\,\mathrm {d} x={\frac {1}{4}}\ln ^{2}(x^{2}+a^{2})}$

${\displaystyle \int \ln \left({\sqrt {x^{n}}}\right)\,\mathrm {d} x={\frac {x}{2}}(\ln(x^{n})-n)}$

${\displaystyle \int \sin(\ln x)\,\mathrm {d} x={\frac {x}{2}}(\sin(\ln x)-\cos(\ln x))}$

${\displaystyle \int \cos(\ln x)\,\mathrm {d} x={\frac {x}{2}}(\sin(\ln x)+\cos(\ln x))}$

${\displaystyle \int e^{x}\left(x\ln x-x-{\frac {1}{x}}\right)\,\mathrm {d} x=e^{x}(x\ln x-x-\ln x)}$

${\displaystyle \int {\frac {1}{e^{x}}}\left({\frac {1}{x}}-\ln x\right)\,\mathrm {d} x={\frac {\ln x}{e^{x}}}}$

${\displaystyle \int (\ln x)^{x}\,\mathrm {d} x=(\ln x)^{x-1}+(\ln(\ln x))(\ln x)^{x}}$

${\displaystyle \int e^{x}\left({\frac {1}{\ln x}}-{\frac {1}{x\ln ^{2}x}}\right)\,\mathrm {d} x={\frac {e^{x}}{\ln x}}}$