Is it acceptable to use Mathematica to derive results in your research?
Is it okay to use Mathematica to solve the integration and use the result in my research?
Of course it is.
It might be useful, though, to make the Mathematica code employed for the calculations available to others, either through a public repository or by a note in the paper suggesting to contact the author(s). This will allow reviewers or other readers to check your calculations, or, possibly, reuse or extend them.
In a paper where I made extensive use of Mathematica, I wrote the following note at the end:
To those interested, the authors can provide the Mathematica notebooks of the full calculations developed in this work.
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.
As Szabolcs writes, verification is important and that's also the case if you were to do computations by hands using methods you think are reliable. There are cases of erroneous results in the peer reviewed literature that were not noticed for quite some time where researches have actually used the wrong formula taken from the published article for many years.
An example is the article by L. Chatterjee, G. Das, R. Goswami: Z. Phys. D 32, 73 (1994), a mistake was made in the computation of the cross section for the radiative capture of a charged particle in the ground state. It was an elementary math mistake due to using inconsistent branch cuts, as pointed out here. That mistake was not noted by the authors precisely because it led to a strange effect of an apparent discontinuity in the cross section, this effect was interpreted as an interesting feature of the capture cross section.