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$\int_{0}^{\,1} {x^p \ln(x)}\, dx$

Find the values of $p$ for which the integral converges.
The integral converges for all values of $p$ in the interval:

For the values of $p$ at which the integral converges, evaluate it.
$\displaystyle{\int_{0}^{\,1} {x^p \ln(x)}\, dx =}$

You can earn partial credit on this problem.