= Solution
For positive bounds, <integration by parts> with $u=\log x$ and $dv=x^2dx$ gives
$$
\boxed{F(a,b)=\left[\frac{x^3}{3}\log x-\frac{x^3}{9}\right]_{a}^{b}.}
$$
For negative bounds, specify the <complex logarithm on the negative real axis>. With argument $\pi$, namely the upper-bank boundary value of the <principal logarithm>, $\log x=\log(-x)+i\pi$. Substitute $t=-x$ in the real part:
$$
\boxed{\int_c^d x^2\log x\,dx=F(-d,-c)+\frac{i\pi}{3}(d^3-c^3).}
$$
More generally, a continuous branch taking argument $(2k+1)\pi$ replaces $i\pi$ by $(2k+1)i\pi$. The imaginary part depends on that choice. The analytic principal branch excludes the negative axis itself, so the boundary-value convention is part of this interpretation.
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