u = z^2, then integrate by parts
Just in case a picture helps (spoiler because of width)...
... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to z, 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).
The general drift is...
Except we also detour through parts...
This spoiler in case of fun...
Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
Balloon Calculus Drawing with LaTeX and Asymptote!