Integrating ln(√x)/x: A Challenging Logarithmic Integral

imull
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Homework Statement


∫(ln(√x))/(x)dx


Homework Equations





The Attempt at a Solution


I am really not sure where to start. All of the other integration problems were relatively simple, sticking with the ∫u'/udu = ln(u).
 
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First extract the square root from the log by remembering the sqrt(x) is the same as x^(1/2).

then see if it becomes more obvious.
 
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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