Find a function that that makes the value of this improper integral equal to 1. I have the following integral:
$$I(t) = \int_{0}^{t} \sqrt{1- \frac{a(x)^2}{c^2}}dx$$
where $a(x)$ is some continuous function of $x$, and $c$ is a constant.  Also $a(x) < c$ for all $x >0$.
It can be seen that the solution to the integral is thus some function of $t$, that is, $I(t)$.
Now I need to find some function $a(x)$, so that $I(t)$ approaches 1 as $t$ approaches $\infty$, but I have no idea where to start.
Does anyone know the solution to this problem, or how to approach it?
Thanks!

$a(x) = c \tanh{(\pi x/2)}$ should do the trick.
