I was constantly losing points on this kind of problem for not changing the limits of integration on substitution steps, but it shouldn't matter since everything gets un-substituted by the end. Also, I looked up the logistic function and verified that I got the correct answer.

That's just MS Word's equation editor, and yes, this is how I do math problems when I have to show the work to anyone but myself. I am quite slow at math, and my handwriting is atrocious, so it's a win-win.