Mathematics

Problem

Let

I(p)=\displaystyle \int_{0}^{1} x^{p}\ln x\,dx

, where

p

is a real parameter. (a) For

p>-1

, describe the behavior of the integrand near

x=0

and explain why the integral may fail to converge for some values of

p

. Your explanation must refer to the relative magnitudes of

x^{p}

and

\ln x

x\to 0^+

. (b) Determine the set of all real numbers

p

for which

I(p)

converges. Provide a clear justification that references an appropriate comparison or limit argument.