Trig function appears awkward to integrate

[Edmundb]

I hope this isn't too simple for the maths forum.

I'm trying to find the length of the locus described by the bottom of a garage door as it is raised. It's all fairly straightforward until I have to integrate a function of the form

√ (sin²θ + k cos²θ)

k does not depend on θ. I tried several approaches to the problem and always came back to the above function.

In the end I integrated numerically and got my number which checks with reality. There's probably some nifty substitution but just I can't find it.

 

[JJacquelin]

See at the elliptic functions.
http://mathworld.wolfram.com/EllipticIntegraloftheSecondKind.html

 

[Edmundb]

Thank you very much indeed. That looks like the clue I need. (As you've probably gathered I'm an Engineer and not a mathematician.)