When you make a substitution of the form $x=asin\theta$ for sines and cosines, shouldn't you check if the values of x on which we're integrating allows for it? Otherwise, if x becomes greater than |x|, wouldn't there be no possible values of $\theta$?
This is how it seems to work in cases like $\sqrt{9-x^2}$, where, it's obvious that you're safe with substituting $x=3 sin\theta$ or $x=3 cos \theta$, since x can't go beyond |3|.
But I haven't heard of any explicit rule about this. Is there one, and if there isn't, why so?
$I=\int_{0}^{2} \sqrt{4-t^2} dt $ is real and $t=2 \sin x$ works
but in $J=\int_{0}^{3} \sqrt{4-t^2} dt $ is imaginary here $t=2 \sin x$ will work again but you will get imaginary value.
Usually, we discard non-real answers.
Simililarly, in indefinite integration $K=\int \sqrt{4-t^2} dt$, we are supposed to avoid the substitution $t=3 \sin x$, because the integrand will become imaginary for $\sin x>2/3.$