Lista de integrais de funcións logarítmicas

A seguinte é unha lista de integrais de funcións logarítmicas.

Nota: neste artigo asúmese que x>0.

$\int \ln xdx=x\ln x-x$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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