Professor couldn't solve this problem and told us we could ask anyone for help solving it for extra credit by the next class. I have tried multiple times and do not think I have it. Answer I have is GmM/2pi integral 0 to 2pi 1/(a+2sqrt(a^2-b^2) cos theta)^2 derivative of theta.

Question

The force of gravitational attraction between the Sun and a planet is F(θ)=GmMr2(θ),F(θ)=GmMr2(θ), where *m* is the mass of the planet, *M* is the mass of the Sun, *G* is a universal constant, and r(θ)r(θ) is the distance between the Sun and the planet when the planet is at an angle *θ* with the major axis of its orbit. Assuming that *M*, *m*, and the ellipse parameters *a* and *b* (half-lengths of the major and minor axes) are given, set up—but do not evaluate—an integral that expresses in terms of G,m,M,a,bG,m,M,a,b the average gravitational force between the Sun and the planet.

Answer 1

The integral of any rational function of trig functions of theta can be turned into an integral of a rational function of z by the u-sub

z = tan(theta/2).

I doubt this is the shortest way in this case, but it is guaranteed to work. Knock yourself out.