How could I evaluate this IMPROPER integra?

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Int from 1 to infinity of ln(X)/x^(3)

--> Any ideas on how to get started(mainly the technique)? thank you.
 
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You would first find an indefinite integral using integration by parts.
 
Let u = ln x and dv/dx = x^-3 and use integration by parts.
And see what happens as the upper bound of the integral tends to infinity.
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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