Find a function that that makes the value of this improper integral equal to 1.

Question

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!

BTW the title of your post is a bit misleading. You are really interested in solving a type of integral equation. — Ron Gordon, Feb 06 '13 at 16:23

score 1 · Accepted Answer · answered Feb 06 '13 at 16:21

1

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

answered Feb 06 '13 at 16:21

Ron Gordon

1 Answers1