In some problems we use reverse substitution for evaluating integrals . For example consider $\int \sqrt{1-x^2}dx$ and $x = \sin(t) \ , \ \frac{-\pi}{2}\le t \le \frac{\pi}{2}$ . In this case , substitution has two properties : $1.$ It covers all of the domain , $2.$ It is a one-to-one function .
I wonder are these conditions always necessary for reverse substitution ? For instance , if we use $x = \sin(t)$ in order to solve $\frac{1}{1-x^2}$ is the answer right (i.e. omitting covering of the whole domain condition) ? Or what would happen if we eliminate the restriction on $t$ (i.e. omitting one-to-one condition) ?
If there is a theorem that determines the conditions for reverse substitution, it would be very helpful .