Computing the difficult integral $\int_0^\infty J_0(x)^4\log(x)dx$

Computing numerically integrals of oscillating functions from $ 0$ to $ \infty$ is a well-known and difficult problem. Here is an example for which I do not know a solution: I know how to compute (to thousands of decimal places if necessary) $ \int_0^\infty J_0(x)^4\,dx$ and $ \int_0^\infty J_0(x)^3\log(x)\,dx$ , where $ J_0$ is the $ J$ -Bessel function. On the other hand, I do not know how to compute $ \int_0^\infty J_0(x)^4\log(x)\,dx$ because the magic of the sumalt program of Pari/GP doesn’t work here because the exponent is even. Note that I do not need this specific integral, my question is more generally how to compute an integral with an even number of $ J$ functions multiplied by some slowly increasing function such as $ \log(x)$ .

Convergence for improper integral $\int_0^\infty x^re^{-x} dx$

I’m trying to find for which values $ r$ the following improper integral converges. $ $ \int_0^\infty x^re^{-x} dx$ $ What I have so far is that $ x^r < e^{\frac{1}{2}x}$ for $ x \geq a$ , which splits the integral into $ $ \int_0^a x^re^{-x} dx + \int_a^\infty e^{-\frac{1}{2}x}$ $ We know the latter interval converges, but I don’t know what to do with the first one.