What does it mean to show that an integral exists?

Question

I am slightly unclear on some terminology that I have come across in the following question:

Let $f(x, t) = xe^{−xt}$.

Show that the integral $I(x) = \int_0^{\infty} f(x, t)\ dt $exists for all $x ≥ 0$.

What does it mean to show that an integral exists? Does this simply mean to show that it converges? I looked this up for some clarification but could only find questions rather than a precise description of what this means.

I would be grateful for any explanation.