INTEGRALS CONTAINING ln(ax)

1.

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

2.

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

3.

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

4.

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

5.

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

6.

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

7.

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

8.

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

9.

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

10.

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

11.

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

12.

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

13.

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

14.

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

15.

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

[Tables]

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