Average Recoil Energy of a Compton Electron

Homework Statement

For an assignment question, I am trying to work out an approximate formula for the average recoil energy of an electron involved in a Compton scattering event (averaged over all scattering angles theta). Note that there are other ways to do this problem than the formula I use below involving the scattering cross sections and whatnot, but we are supposed to use the formula below and then compare our answer to the cross section method to see how they differ.

Homework Equations

We are to use the equation for the recoil energy of a Compton electron as a function of the photon scattering angle theta:

$$\frac{E_r_e_c_o_i_l}{h\nu} = \frac{\epsilon(1-cos\theta)}{1+\epsilon(1-cos\theta)}$$

I want to find

$$\frac{\bar{E_r_e_c_o_i_l}}{h\nu}$$

I.e - the energy of the recoil electron (normalised to the incoming photon energy) averaged over all possible scattering angles

The Attempt at a Solution

$$\frac{\bar{E_r_e_c_o_i_l}}{h\nu} = \frac{\displaystyle\int^\pi_0 F(\theta)\,d\theta}{\displaystyle\int^\pi_0 \,d\theta}$$

Where

$$F(\theta) = \frac{\epsilon(1-cos\theta)}{1+\epsilon(1-cos\theta)}$$

The problem is, I have absolutely no idea how to even start doing this integral! A substitution? Integration by parts? Manipulation using trig identities, and then one or both of the above?

Any guidance or direction from a more mathematically able mind than my own would be much appreciated... This problem is killing me.

$$\int d\Omega = \int \int sin(\theta) d\theta d\phi$$