(번역) List of integrals of logarithmic functions

Original article: w:List of integrals of logarithmic functions

 

다음은 로그 함수(logarithmic function)의 적분(integral) (역도함수(antiderivative))의 목록입니다. 적분 함수의 전체 목록에 대해, 적분의 목록(list of integrals)을 참조하십시오.

주목: x > 0이 이 기사의 전체에 가정되고, 적분화의 상수(constant of integration)는 간결성을 위해 생략됩니다.

Integrals involving only logarithmic functions

${\displaystyle \int {\log }_{a}xdx=x{\log }_{a}x-\frac{x}{\ln a}=\frac{x\ln x-x}{\ln a}}$

${\displaystyle \int \ln (ax)dx=x\ln (ax)-x}$

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

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

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

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

${\displaystyle \int \frac{dx}{\ln x}=\mathrm{li}(x)}$, the logarithmic integral.

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

Integrals involving logarithmic and power functions

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

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

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

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

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

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

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

${\displaystyle \int \frac{dx}{x\ln x\ln \ln x}=\ln |\ln |\ln x||}$, etc.

${\displaystyle \int \frac{dx}{x\ln \ln x}=\mathrm{li}(\ln x)}$

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

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

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

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

Integrals involving logarithmic and trigonometric functions

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

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

Integrals involving logarithmic and exponential functions

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

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

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

n consecutive integrations

${\displaystyle n}$ 연속적인 적분화에 대해, 다음 공식은

${\displaystyle \int \ln xdx=x(\ln x-1)+C_0}$

다음으로 일반화됩니다:

${\displaystyle \int \cdots \int \ln xdx\cdots dx=\frac{x^n}{n!}(\ln x-\sum _{k=1}^n\frac{1}{k})+\sum _{k=0}^{n-1}{C}_k\frac{x^k}{k!}}$