Improper integral interval

Question

For which values of $p$ does the integral $$\int_{1}^{\infty} x^{p} \ln(x)\,dx$$ converge?

I've graphed the function for values of $p$ less than and greater than $1$ to get an idea of the function, when $p = 1$ the graph diverges and when $p = -1$ the graph converges on to $0$, but I'm not sure what my interval would be.

Answer 1

Integrate by parts:

$$\int x^p\ln(x)~\mathrm dx=\frac1{p+1}\left[x^{p+1}\ln(x)-\int x^p~\mathrm dx\right]$$

Now it should be obvious when it converges/diverges.