Thank you for the opportunity to help you with your question!

How the fudge am I supposed to do this without parts? Is there supposed to be some kind of antiderivative to this function with an x in front of it or something? This is like three functions in one.if you may have reached the right integral, which is

∫102πxe−x2dx.

You can quickly integrate by making the substitution u=x2. Please let me know if you need any clarification. I'm always happy to answer your questions.