determining the limits in $u$-substitution

Question

I have a fundamental question regarding $u$-substitution: Suppose that want to calculate the integral $$ \int_{a}^{b}f(x)\,dx $$ and we wish to use a $u$-substitution of the form $x=g(u)$ (rather then $u=g(x)$) and suppose that we can express $f(x)\,dx$ using $u$ and $du$. How to choose the limits of the new integral? can we choose any $\alpha$ and $\beta$ such that $g(\alpha)=a$ and $g(\beta)=b$, or should we choose $\alpha$ and $\beta$ such that $g(x)$ is invertible in $[\alpha,\beta]$? for example, suppose we want to calculate $$ \int_{0}^{1/2}x^2\,dx $$ by using the subsition $x=\sin u$. Here $x^2=\sin^2x$, so $2x\,dx=2\sin u\cos u\,du$, that is $x\,dx=\sin u\cos u\,du$. Since $x=\sin u$, it follows that $x^2\,dx=\sin^2u\cos u\,du$. Therefore, our integral is $$ \int_{?}^{?}\sin^2u\cos u\,du $$ Now, in this example we can choose any limits $\alpha$ and $\beta$ such that $\sin\alpha=0$ and $\sin\beta=1/2$ (such as $\alpha=-\pi$ and $\beta=\pi/6$). Does that work in general? Note that I deliberately took a non invertible substitution. If it does not true I will be happy to see some example. Thanks a lot!