For what $p$ is $\frac{1}{(x(1+\ln(x)^2))^p}$ Lebesgue integrable?

Question

I'm trying to use the fact that given $f:[a,\infty)\to\mathbb{R}$ Riemann integrable for every closed interval $[c,d]\subset [a,\infty)$, then $f$ is Lebesgue integrable if, and only if, $\int_a^\infty|f(x)| \, dx$ exists.

In particular, $f(x)=\frac{1}{x(1+\ln(x)^2)}$ is $p$-Lebesgue integrable if $\int_0^\infty\frac{1}{x^p(1+\ln(x)^2)^p} \, dx<\infty $. Here using the fact that $f>0$.

But I can't solve this Riemann integral.

Answer 1

Your function produces infinities in both endpoints of the interval. The usual approach is to split the integral using some well-behaved intermediate point, say $x=1$. Therefore, $\int \limits _0 ^\infty = \int \limits _0 ^1 + \int \limits _1 ^\infty$.

Concerning the first one, make the change of variable $x = \Bbb e ^{-t}$, obtaining $\int \limits _0 ^\infty \frac {\Bbb e ^{(p-1)t}} {(1+t^2)^p} \Bbb d t$. This clearly converges if $p \le 1$. If $p>1$, the function is essentially unbounded towards $\infty$, so no chance of being integrable.

Concerning the second one, the same change of variable will produce $\int \limits _{-\infty} ^0 \frac {\Bbb e ^{(p-1)t}} {(1+t^2)^p} \Bbb d t$ which, with exactly the same argument as above, converges only for $p \ge 1$.

To conclude, when $p>1$, the first part diverges and the second one converges, so the whole integral diverges. When $p<1$ it diverges again. The only case of convergence is $p=1$.

Answer 2

Using the inequality $\frac{x}{x+1} < \log (1+x) < x$ for ll $x>-1$ and $x \ne 0$. Let $x=u-1$ then the inequality becomes $\frac{u-1}{u} < \log (u) < u-1$ for all $u>0$. Thus, $\frac{(x-1)^2}{x^2}< (\ln x)^2<(x-1)^2$ so that $1+\frac{(x-1)^2}{x^2}<1+ (\ln x)^2<1+(x-1)^2$ which implies that $\frac{1}{1+\frac{(x-1)^2}{x^2}}>\frac{1}{1+ (\ln x)^2}>\frac{1}{1+(x-1)^2}$.

Thus, $\frac{x^{2p}}{(x^2+(x-1)^2)^p}>\frac{1}{(1+ (\ln x)^2)^p}>\frac{1}{(1+(x-1)^2)^p}$.

Alos, since $x>0$ then, $$\frac{1}{x^p(1+ (\ln x)^2)^p}<\frac{x^{p}}{(x^2+(x-1)^2)^p}.$$

Using Maple 12, I tried to integrate $\int_0^{\infty}{\frac{x^{p}}{(x^2+(x-1)^2)^p}dx}$ I get that the integral converges when $p\ge2$, diverges when $p<2$.