Dan Stanger wrote:
> Integrals like this can often be done more easily with the Euler
> substitution, here is a link: http://eom.springer.de/e/e036590.htm .
I have looked at the springer link. So this substitution boils down to the
fact that every degree 2 curve is of genus 0 and thus has rational
parametrization. It is another instance of the fact that any function
rational in sin and cos can be reduced to purely rational by using
the tangent of the half angle, as we discussed recently.
There also exists an Euler trick for genus 1 curves (cubics) which
has to do with the natural addition on the cubic, and also with square roots
but i don't remember exactly what it is. There is a big discussion of this
trick in a paper by van Moerbeke and Horozov called "The full Geometry of
Kowalewski's top and (1, 2)-abelian surfaces".
--
Michel Talon