I have a somewhat general question, but have trouble formulating it I will illustrate my question with an example, in the hope that someone can generalise this.
Consider:
$$I = \int_{0}^ \infty \frac{dx}{1+(2x+1)^2}$$
Let $t = 2x+1, dt/2 = dx$. If $x = 0$, then $t = 1$ and if $x = \infty$ then $t = \infty$. Hence
$$I = 1/2\int_{1}^\infty \frac{dt}{1+t^2} = 1/2 (\lim_{t \to \infty} \arctan t - \pi/4) = 1/2( \pi/2 - \pi/4) = \pi/8$$
I want to formalize the "$x = \infty$, then $t = \infty$" part.
It seems that we can do:
$$I = \lim_{s \to \infty} \int_0^s \frac{dx}{1 +(2x+1)^2} = 1/2\lim_{s} \int_1^{2s+1} \frac{dt}{1+t^2} $$
$$= 1/2(\lim_s \arctan(2s+1) - \pi/4) = \pi/8$$
Is there a general substitution rule for improper integrals of this kind?