How do you integrate this?
ie the bottom is to the power of 3/2
Just in case a picture helps...
... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to the main variable (y on the left, theta after substituting), and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).
And I've used theta instead of u because that's usual (despite my earlier choice of u!) when you use a trig substitution to view the original function as though it were the OUTER part of the result of a chain rule differentiation. Hope the pic helps to get a handle on the logic. I'm sure others will lend further (and more conventional) advice...
The general drift is...
And the rest...
Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
Balloon Calculus Drawing with LaTeX and Asymptote!