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Why is this integral divergent?
Given $f(x) = x^4 \ln(x)$ if $x>0$ and $g(x) = -x^2$ if $x\leq 0$ then calculate:
$$\int_0^\infty x^4\ln(x)\,\mathrm dx+\int_0^\infty -x^2\,\mathrm dx$$
I am not sure how to set this up using traditional methods so I tried instead to calculate the difference between the two integrals which I then took the limit as $x\to\infty$ of this difference and found that I get $\infty$ which means that the integral diverges. But why is that since $f(x)$ and $g(x)$ are non-negative integrands and this integral is just a weighted sum of these?
Also, this is not homework, just a question which I stumbled upon while looking for different approaches to some questions in my textbook.
EDIT: I checked my work and apparently I’m wrong, my mistake was that I was assuming continuity at $x=0$ which is not correct, $g(x)$ should not be continuous at $x=0$ if $g(0)=0$. The correct way to calculate this is by adding the limit as $x \to 0^+$ with $x>0$ and the limit as $x \to 0^-$ with
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