Convergence and Divergence of Improper Integral $\int_{0}^{\infty}x^m ( \ln x)^n dx$

Question

Is this integral $\int_{0}^{\infty}x^m (lnx)^n dx$ divergent or convergent? Why? I understand that if we plot the integrand at any values of m,n (except zero), we can clearly see that the limit does not exist, however how can I prove this mathematically, if possible?

Answer 1

For $m = n = 1$ we have

$$\int_0^{\infty}x \ln x \text{ d} x = \left[{x^2\over 4}(2 \ln x -1)\right]_0^{\infty} = \lim_{x\to \infty}{x^2\over 4}(2 \ln x -1) \to \infty$$

Since we disproved convergence for a particular case, the integral cannot converge for the general case either.