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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.

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    $\begingroup$ What are you asking? The integral converges for $p<-1$ and diverges otherwise. $\endgroup$ – Mark Viola May 29 '17 at 14:38
  • $\begingroup$ im looking for the interval that the function converges for p $\endgroup$ – guy_sensei May 29 '17 at 14:42
  • $\begingroup$ And the answer is that the integral converges for $p<-1$ and diverges otherwise. $\endgroup$ – Mark Viola May 29 '17 at 14:44
  • $\begingroup$ The question shows the integral above and asks "the integral converges for p in { }" ; State your answer as an interval. $\endgroup$ – guy_sensei May 29 '17 at 14:50
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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.

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  • $\begingroup$ (+1) ... but inasmuch as I gave the answer in a comment, and the OP demanded "State your answer as an interval," it might not be obvious to the OP. $\endgroup$ – Mark Viola May 29 '17 at 14:54
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    $\begingroup$ @MarkViola They accepted this, so it should be all good. $\endgroup$ – Simply Beautiful Art May 29 '17 at 15:17

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