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(and i'm still noodling over your stupid distance-from-point-on-x-axis-to-point-on-unit-circle integral; not sure if you think of it that way but that's one of the things it is)
yep 1 - 2*a*cos(θ) + a^2 = sin^2(θ) + cos^2(θ) - 2*a*cos(θ) + a^2 = sin^2(θ) + [a - cos(θ)]^2 = the square of the distance from (a, 0) to (cos(θ), sin(θ)) maybe that has nothing to do with how to solve it but it's what immediately came to mind
Neat! If you could solve it that way it'd be pretty cool
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