Restriction caused by substitution We often come across integrals where we use various substitutions to reduce our problem. A common one is the trigonometric substitution.
When we substitute $x=\sin \theta$,  aren't we restricting $x$'s values to $[-1,1]$?
Here's a quick example:
$$\int_2^3 xdx$$
In this case, $x$ is always in $[2,3]$. So is $x=\sin \theta$ a valid substitution here?
In general, I would like to say that a substitution in an indefinite integration is valid, if and only if the variable substituted has the same range as the new variable.
 A: You 'run' a definite integral over a set. In a way you can think of a ' definite integral' as a procedure which takes in a subset(S) of $\mathrm{R}$ and tells the area of a function underneath the curve and x-axis in that interval.

When you do a change of variables, since you are changing the function you are integrating, you must also change the set you are integrating over. If you have are integrating over a set $[a,b]$ and you do a substitution of the form below:
$$ x= g(t)$$
Then, in the new domain of $t$, your domain is transformed like so:
$$ [a,b] \to [ g^{-1} (a) , g^{-1} (b) ] $$

You can think that as we vary 't', we vary $g(t)$ and this, in turn, varies the output. And, so, yes we can only do a substitution if the $g(t)$ can span the original domain which we were integrating over.
Given the above, if you are doing a change of variables, first of all, you need to make sure there is a well-defined inverse function of the new function you are integrating such that you can change your bounds properly.
In the particular case, it seems we can't find $ \sin^{-1} (2)$ or $ \sin^{-1} (3)$ whilst constrained to the set of reals. And hence, that substitution is not feasible in this case.
