I gave a rather unclear explanation of the following in a lecture a couple of days ago and promised to (at least attempt to) clarify a bit.

You know that:


\int_0^{10}\frac{1}{x} dx


is divergent and gives an answer of \infty. If you have an integral of the form:




as we had in class, you don’t need to calculate this separately if you already know about the behaviour of 1/x about x=0 as they are exactly the same thing, just shifted (at least their convergence properties are the same).


The take-home message is that if you have:


\int_a^b \frac{c}{(x-a)^p}dx


and you already know the behaviour of:


\int_0^d \frac{1}{x^p}dx


then the integral you are interested in will have the same convergence or divergence behaviour.

How clear is this post?