I am not going to do the definite integral, rather I will find the anti-derivative and leave you to evaluate them at the endpoints.
Now, the concepts of differenencials, splitting the dy and the dx as if they are fractions.
Like for example,
Is not really acceptable among mathemations because they are not fractions. Thus, I use a more formal approach which does not involve splitting the denominator of the differencial.
I wrote a Calculus thread over here.
(The math program that is used to generate math code is math working properly now, thus you might not see everything).
In that thread I explain how I do integration via the substitution rule my way.
And yes, that is my own techinque.