How to say when the integral converge and when diverge?

Question

I have the following integral.

$$I=\int_a^b |x|^{-p} dx$$ where $a<b$ are finite real numbers and $p\leq 0$ is a non-negative real number. If we start solving the integral we will come up the following result,

$$I=\frac{|x|^{-p+1}}{-p+1}|_{a}^{b}$$.

Where I stuck is that how one can say for which value of a, b and p the above integral converge and diverge? Any detailed suggestion will be highly appreciated.

Answer 1

If $-p>0$ there isn't problems. If $-p<0$, then the integrand function is $$\frac{1}{|x|^p}$$ this integral doesn't diverge if $|x|\neq 0$ in $[a,b]$.

Answer 2

First one should note that if $p=1$, the $I = \ln(x)|^b_a$, and I guess you mean $p\ge 0$ is a non-negative number. In this case, if $0<a,b<\infty$, then the integral will converge. The problem comes in if either one of $a,b=0$.