Zerrenda:Funtzio logaritmikoen integralak

Ondorengoa funtzio logaritmikoen integralen zerrenda bat da (jatorrizkoak edo antideribatuak). Integralen zerrenda osatuago nahi baduzu, ikusi integralen zerrenda.

Oharra: Artikulu honetan x>0 da.

$\int \ln axdx=x\ln ax-x+K$

$\int \ln (ax+b)dx=\frac{(ax+b)\ln (ax+b)-(ax+b)}{a}+K$

$\int (\ln x{)}^{2}dx=x(\ln x{)}^2-2x\ln x+2x+K$

$\int (\ln x{)}^{n}dx=x\sum _{k=0}^n(-1{)}^{n-k}\frac{n!}{k!}(\ln x{)}^k+K$

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

$\int \frac{dx}{(\ln x{)}^n}=-\frac{x}{(n-1)(\ln x{)}^{n-1}}+\frac{1}{n-1}\int \frac{dx}{(\ln x{)}^{n-1}}\text{( }n\ne 1\text{)}$

$\int x^m\ln xdx=x^{m+1}(\frac{\ln x}{m+1}-\frac{1}{(m+1{)}^2})+K\text{( }m\ne -1\text{)}$

$\int x^m(\ln x{)}^{n}dx=\frac{x^{m+1}(\ln x{)}^n}{m+1}-\frac{n}{m+1}\int x^m(\ln x{)}^{n-1}dx\text{( }m\ne -1\text{)}$

$\int \frac{(\ln x{)}^{n}dx}{x}=\frac{(\ln x{)}^{n+1}}{n+1}+K\text{( }n\ne -1\text{)}$

$\int \frac{\ln x^{n}dx}{x}=\frac{(\ln x^n{)}^2}{2n}+K\text{( }n\ne 0\text{)}$

$\int \frac{\ln xdx}{x^m}=-\frac{\ln x}{(m-1)x^{m-1}}-\frac{1}{(m-1{)}^2{x}^{m-1}}+K\text{( }m\ne 1\text{)}$

$\int \frac{(\ln x{)}^{n}dx}{x^m}=-\frac{(\ln x{)}^n}{(m-1)x^{m-1}}+\frac{n}{m-1}\int \frac{(\ln x{)}^{n-1}dx}{x^m}\text{( }m\ne 1\text{)}$

$\int \frac{x^{m}dx}{(\ln x{)}^n}=-\frac{x^{m+1}}{(n-1)(\ln x{)}^{n-1}}+\frac{m+1}{n-1}\int \frac{x^{m}dx}{(\ln x{)}^{n-1}}\text{( }n\ne 1\text{)}$

$\int \frac{dx}{x\ln x}=\ln |\ln x|+K$

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

$\int \frac{dx}{x(\ln x{)}^n}=-\frac{1}{(n-1)(\ln x{)}^{n-1}}+K\text{( }n\ne 1\text{)}$

$\int \ln (x^2+a^2)dx=x\ln (x^2+a^2)-2x+2a{\tan }^{-1}\frac{x}{a}+K$

$\int \frac{x}{x^2+a^2}\ln (x^2+a^2)dx=\frac{1}{4}{\ln }^2(x^2+a^2)+K$

$\int \sin (\ln x)dx=\frac{x}{2}(\sin (\ln x)-\cos (\ln x))+K$

$\int \cos (\ln x)dx=\frac{x}{2}(\sin (\ln x)+\cos (\ln x))+K$

$\int e^x(x\ln x-x-\frac{1}{x})dx=e^x(x\ln x-x-\ln x)+K$

$\int \frac{1}{e^x}(\frac{1}{x}-\ln x)dx=\frac{\ln x}{e^x}+K$

$\int (\ln x{)}^{x}dx=(\ln x{)}^{x-1}+(\ln (\ln x))(\ln x{)}^x+K$

$\int e^x(\frac{1}{\ln x}-\frac{1}{x{\ln }^{2}x})dx=\frac{e^x}{\ln x}+K$

Bibliografia

• Milton Abramowitz eta Irene A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 1964. Liburu klasiko honetan integral batzuen zerrenda dago 69. or. .