I have an integral that involves a modulus term under the integral sign, that I'm not too sure how to deal with.
$$\int_{|z|=R} \dfrac{|dz|}{|z-c|^4}, R > 0, |c| \neq R$$
OK, so now suppose I know how to deal with the $|dz|$ term. I still don't know what to do with the term on the bottom. I could parameterize $z(\theta) = Re^{i\theta}$, then $z'(\theta) = iRe^{i\theta}$, and $|z'(\theta)| = R$, so
$$\int_{|z|=R} \dfrac{|dz|}{|z-c|^4} = \int_{|z|=R} \dfrac{1}{|z(\theta)-c|^4} |z'(\theta)| d\theta = R\int_{0}^{2\pi} \dfrac{1}{|Re^{i\theta}-c|^4} d\theta$$
And now I'm not sure how to proceed. This doesn't seem to fit a Cauchy-esque theorem that would give me that this is multiple of $2\pi i$, or something to that effect.