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.