Answer
$-\dfrac{1}{4}$
Work Step by Step
Re-arrange the given integral as follows:
$\int_{0}^{1} x \ln x dx=[(1/2) (\ln x-(1/2)]_0^1$ ....(1)
Then, we have
$[(1/2) (\ln x-(1/2)]_0^1=(\dfrac{1}{2}) (1)^2 (\ln 1-\ln (1/2))-(\dfrac{1}{2}) (0)^2 (-\infty-\dfrac{1}{2})$
Thus, equation (1) becomes
$\dfrac{1}{2}(0-\dfrac{1}{2})=-\dfrac{1}{4}$
