1.52 Integrals of unbounded functions

We can integrate continuous functions on $[a,b]$ by the approximation procedure of Chapter 5 from MATH101 Calculus. Now we consider functions that are continuous except at one of the end-points. First, suppose that $f$ is continuous on $(a,b]$ but discontinuous at $a$ (possibly $f$ is unbounded at $a$). Let $\delta$ (small Greek delta) stand for a small positive number.

Then for $\delta>0$ we can integrate over $(a+\delta,b)$ and form $\int_{a+\delta}^{b}f(x)\,dx.$ We can define the improper integral of $f$ over $(a,b)$ to be

where this limit exists; otherwise, we say that the integral $\int_{a}^{b}f(x)\,dx$ diverges.