TITLE: What does it mean to show that an integral exists?
QUESTION [6 upvotes]: 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.

REPLY [4 votes]: The (improper Riemann) integral $\int \limits_0^{\infty} g(t) \, \text{d} t$ is said to exist if:

For each $a > 0$ the proper Riemann integral $\int \limits_0^a g(t) \, \text{d} t$ exists, i.e. $g$ is Riemann integrable over $[0, a]$;
The limit $\lim \limits_{a \to \infty} \int \limits_0^a g(t) \, \text{d} t$ exists.

The integral $\int \limits_0^{\infty} g(t) \, \text{d} t$ is then the value of the limit. Now for your question just apply the definition to $g(t) := f(x, t)$, where $x \geqslant 0$ is fixed.