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)$ .

# Tag: $\int_0^{\infty}

## \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})}$ .