INTEGRALS CONTAINING ln(ax)

1.

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

2.

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

3.

$\displaystyle\int x^m\ln xdx=\displaystyle \frac{x^{m+1}}{m+1}\left(\ln x-\displaystyle \frac{1}{m+1}\right)$

4.

$\displaystyle\int\displaystyle \frac{\ln x}{x}dx=\displaystyle \frac{1}{2}\ln^2 x$

5.

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

6.

$\displaystyle\int\ln^2 xdx=x\ln^2 x-2x\ln x+2x$

7.

$\displaystyle\int\displaystyle \frac{\ln^n xdx}{x}=\displaystyle \frac{\ln^{n+1}x}{n+1}$

8.

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

9.

$\displaystyle\int\displaystyle \frac{dx}{\ln x}=\ln (\ln x)+\ln x+\displaystyle \frac{\ln^2 x}{2\cdot 2!}+\displaystyle \frac{\ln^3 x}{3\cdot 3!}+\cdot\cdot\cdot$

10.

$\displaystyle\int\displaystyle \frac{x^m dx}{\ln x}=\ln (\ln x)+(m+1)\ln x + \d... ...n^2 x}{2\cdot 2!}+\displaystyle \frac{(m+1)^3\ln^3x}{3\cdot 3!}+\cdot\cdot\cdot$

11.

$\displaystyle\int\ln^n xdx=x\ln^n x-n\int\ln^{n-1}xdx$

12.

$\displaystyle\int x^m\ln^n xdx=\displaystyle \frac{x^{m+1}\ln^n x}{m+1}-\displaystyle \frac{n}{m+1}\int x^m\ln^{n-1}xdx$

13.

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

14.

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

15.

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