Here is a problem that I was wondering if anyone could help me out on.

An ellipse has parametric equations

,

for

where

.

Show that its perimeter is given by the integral

.

Hence show that, if the ellipse is close to circular, the perimeter is approximately equal to

I managed to show that the perimeter is given by the given integral but I cannot make headway with the approximation. I rearrange the integral to give this;

I thought maybe I could integrate the binomial for the terms under the root symbol (using the maclaurin expansion of

but then that would take me to an answer with higher powers of

so I don't really know where to go with it.

Any thoughts would be appreciated!