I think you can treat it as:
EDIT: After you edited your post, yes, that's what I would have done
Sorry, I have edited my first post without actually taking a look at whether the integral was right or wrong.
My second post says it.
In fact, it's only my edit in my first post which is contradicting. The 'formula' that I posted remains the same and applies.
So, your integral is wrong. You need to put the differential of the function as denominator, not its integral.
If you had to differentiate
This becomes:
When you integrate, you divide instead of multiplying the differential of the function.