# 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!

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BTW the title of your post is a bit misleading. You are really interested in solving a type of integral equation. –  Ron Gordon Feb 6 '13 at 16:23
I wasn't sure about the title. Please feel free to correct. –  Mew Feb 6 '13 at 16:24

## 1 Answer

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

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Thanks for your answer. –  Mew Feb 6 '13 at 16:28
NP, glad I could help. –  Ron Gordon Feb 6 '13 at 16:28