# Thread: Help with an integral

1. ## Help with an integral

Hi I'm doing a question regarding averaging a radial perterbation about an orbit.

I've reduced the problem to needing to show that
$\int_0^{2\pi} \frac{\sin f}{(1+e\cos f)^{\gamma +2}} df = 0$

Where we have that $0 could I have a hint from anyone as to how to proceed with this?

2. ## Re: Help with an integral

Originally Posted by thelostchild
Hi I'm doing a question regarding averaging a radial perterbation about an orbit.

I've reduced the problem to needing to show that
$\int_0^{2\pi} \frac{\sin f}{(1+e\cos f)^{\gamma +2}} df = 0$

Where we have that $0 could I have a hint from anyone as to how to proceed with this?
$u = 1+e\cos{f}$

$du = -e\sin{f} \, df$

substitute and reset limits of integration ...

$-\frac{1}{e}\int_{1+e}^{1+e} \frac{du}{u^{\gamma + 2}}$

note that for any function $f$ , $\int_a^a f(x) \, dx = 0$

3. ## Re: Help with an integral

I'm afraid that method doesn't work, as the integral is non convergent for -1 < Re(e) < 1 or Im(e) != 0 so there is obviously something fishy with that substitution

I have now figured it out, to do it (semi) rigorously, you taylor expand the (1+e*cos(f))^(-gamma-2) term which is a convergent series for -1<e<1 and integrate term by term, all of which evaluate to zero.

Thanks for the help though