For the function
$$S(x) = \int_{0}^{x} sin(t^2) dt$$
If we apply the change of variables $u=t^2$, $$S(x) = \int_{0}^{f(x)} sin(u)h(u) du$$
what are the functions $f(x)$ and $h(u)$?
And, if we set $$Z(y) = \int_{0}^{y} sin(u)h(u) du$$, how do we find $Z'(y)$ using the fundamental theorem of calculus and the chain rule if we have $$Z(y)=S(g(y))$$
I tried $u=t^2$ so $du=2tdt$ and $$S(x) = \int_{0}^{x} sin(t^2) dt=\int_{0}^{x^2}sin(u)\frac{1}{2t}du$$ So I think $f(x)=x^2$ and $h(u)=\frac{1}{2\sqrt{u}}$ but I am not sure. How would I proceed from here?
