## 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 $$\log(x)$$.

## \int_0^\infty \sin{\left(\frac{1}{4x^2}\right)}\frac{\ln{x}}{x^2}dx

Hi I want to solve the following Integral but failed $$\int_0^\infty \sin{\left(\frac{1}{4x^2}\right)}\frac{\ln{x}}{x^2}dx$$

I wonder if you guys can help

## Evaluating $\int_{0}^{\infty} \frac{1-e^{-\alpha x^2}}{x^2} dx$

I need help with following integral: $$\int_{0}^{\infty} \frac{1-e^{-\alpha x^2}}{x^2} dx, \alpha>0$$

## 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.

## What is $\int_0^{\infty} \frac{e^{ax}-e^{bx}}{(1+e^{ax})(1+e^{bx})}$?

Calculate $$\int_0^{\infty} \frac{e^{ax}-e^{bx}}{(1+e^{ax})(1+e^{bx})}$$.