Subject: Calculus

List Of Integrals Of Logarithmic Functions

Some Common Integrals Involving Logarithmic Functions
\int \ln{|bx|}dx = x \ln{|bx|} - x + C  \int \ln{|bx + c|}dx = (bx + c) \ln{|bx + c|} - (bx + c) \over b + C
    
\int (\ln{x})^2dx = x (\ln{x})^2- 2x\ln{x} + 2x + C  \int (\ln{x})^ndx = x \sum_{k=0}^{n}(-1)^{[n-k]} \frac{n!}{k!}\ln{x}^k
    
\int \frac{dx}{\ln{x}} = \ln{|\ln{x}|} + \ln{x} + \sum_{k=2}^{\infty}\frac{\ln{x}^k}{k\cdot k!}   \int \frac{dx}{\ln{x}^{n}} = -x \over (n-1)\ln{x}^{n-1} + \frac{1}{n-1}\int \frac{dx}{\ln{x}^{n-1}}
    
\int x^n \ln{x}dx = \frac{x^{n+1}}{n+1} \left (\ln{x} - \frac{1}{n+1}\right ) + C   \int x^n \ln{x}^rdx = \frac{x^{n+1}}{n+1}\ln{x}^{r} - \frac{r}{n+1}\int x^n \ln{x}^{r-1}dx
    
\int \frac{(\ln{x})^r}{x}dx = (\ln{x})^{r+1}\over r+1 + C   \int \frac{\ln{x}^r}{x}dx = (\ln{x}^{r})^2 \over 2r + C
    
\int \frac{\ln{x}}{x^r}dx = -\ln{x}\over (m-1)x^{m-1} + -1\over (m-1)^2 x^{m-1} +C   \int \frac{\ln{x}^r}{x^m}dx = -\ln{x}^r\over (m-1)x^{m-1} + \frac{r}{m-1}\int \frac{(\ln{x})^{r-1}}{x^m}dx
    
\int \frac{dx}{x\ln{x}}dx = \ln{|\ln{x}|} + C   \int \frac{dx}{x(\ln{x})^r}dx = - 1 \over (r-1)(\ln{x})^{r-1} + C
    
\int \sin{\ln{x}}dx = \frac{x}{2}\left (\sin{\ln{x}}-\cos{\ln{x}} \right )  \int \cos{\ln{x}}dx = \frac{x}{2}\left (\sin{\ln{x}} + \cos{\ln{x}} \right )