Yes, that's correct, but there are a few mistakes that I feel the need to point out.
This should be by the chain rule.Using and .
This should be
I'm not quite sure if this is a typo or not but you have to either re-write your integral after you've integrated in terms of again if you're going to evaluate it with those limits (which I think you've done by putting the back in but it got a bit confusing by keeping it in terms of ) OR change your limits for i.e. ..