Question about asympotic expansion of $\int_0^x t\sqrt{ln(t)} dt$

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Question: Find the constants $$a_0, a_1, a_2$$ in the asympotic expansion

$$\int_0^x t\sqrt{ln(t)} dt$$ = $a_0(x^2)(lnx)^\frac 12$ + $a_1\frac {x^2}{(lnx)^\frac 12}$ + $a_2\frac {x^2}{(lnx)^\frac 32}$ + ...

In what regime is this a useful asymptotic expansion? Justify your answer.

I'm confused about this last part, what does "In what regime" mean. Someone please help! Thank you!!