I take this question as not specific to Mathematica, but equally relevant to any other computer algebra system.
You have an integral or an equation that you cannot solve. You have a piece of software that will give you a result. But you don't know how it arrived to the result. Is it okay to use it?
What matters is whether the result is correct, not how you arrived to it. You should understand the problem you are solving, and you should verify the solution.
Personally, I would be very uncomfortable using such a result blindly, especially knowing how easily certain automated symbolic calculations, such as definite integration, can go wrong. But luckily, most of these types of results are much easier to verify than to compute. You have an indefinite integral? Differentiate it! An equation? Substitute back the solution! A definite integral? Do it numerically and compare to the symbolic solution!
Writing in your paper that "this is the result of the integral because Mathematica said so" is not okay, if you didn't verify it. Just stating the result without saying how you arrived to it is fine for as long as you have verified it and it is also obvious enough for any reader how to verify it. If it is not obvious, then prove the result in the paper, i.e. show how you verified it.
Given that you mention integration, I should point out that doing definite integrals automatically is notoriously difficult, and all computer algebra systems will occasionally return wrong results. That's a very good reason to always verify.