I'm trying to calculate an integral which looks unfortunately divergent. The structure is similar to a loop integral but the appendix in the Peskin Quantum Field Theory textbook didn't have a useable form. The integral form is (I did a u substitution to make it easier to look at)
$$ \int_x^{\infty}du \frac{u^2}{\omega - u} $$
Does anyone know of a workable form for this? Introducing a cutoff is possible but I would prefer not to.
Thank you!
The degree of the numerator is greater than the degree of the denominator, which is a signal to do some old-fashioned long division of polynomials. When you do, you get a quotient of $-u-\omega$ and a remainder of $\omega^2$. This means that $$ \frac{u^2}{\omega-u} = -u-\omega +\frac{\omega^2}{\omega-u.} $$ Now the integral is doable, the last term giving you a logarithm (but the $-u$ term is already a problem as $u \to \infty$). Can you handle the rest?
After the fact: looking at this some more, you may need to be really careful about what $x$ is, making sure that everything makes sense here and is set up correctly. Again, I'll leave that to you. So, take my answer as how to get the general antiderivative.