Lebesgue integrability of $f_n (x) := \begin{cases} \frac{\lvert\ln (n^7x^2)\rvert}{3+n^4x^2} & x \ne 0 \\ 1 & x=0\end{cases}$ on $\mathbb R$

Question

It is a part of an exercise. The functions of the title are from $\mathbb R$ to $\mathbb R$. How can I see if they are integrable with respect to unidimensional Lebesgue measure?

I tried Hölder inequality but nothing, since one of the factors is $+\infty$. The hope is to find one $g \in L^1 \mathbb R$ such that $\lvert f \rvert \le g$, but it seems to require some finer work.

How would you tackle that?

Answer 1

Using the fact that the function is symmetric about $x=0$, we can compute \begin{align} \int |f_n(x)|dx &= \int \frac{|7\ln n+2\ln |x||}{3+n^4x^2}dx \\ &< 14\ln n \int_0^\infty \frac{1}{3+x^2}dx + \frac{4}{3}\int_0^1 -\ln xdx + 4\int_1^\infty \frac{\ln x}{x^2}dx\\ &=\frac{7\ln n}{\sqrt{3}}\pi+\frac{4}{3}+4 <\infty. \end{align}