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Improper integrals
Imagine you want to find the area under the function $f(x)=\frac{1}{x^2}$ from $x=1$ all the way to infinity $x=\infty$. This kind of calculation is known as an improper integral since one of the endpoints of the integration is not a regular number, but infinity.
Nevertheless we can compute this integral: \[ \int_1^\infty \frac{1}{x^2}\;dx \equiv \lim_{b\to\infty} \int_1^b\frac{1}{x^2}\; dx = \lim_{b\to\infty} \left[ \frac{-1}{x} \right]_1^b = \lim_{b\to\infty} \left[-\frac{1}{b} + \frac{1}{1}\right] = 1. \]
Definitions
An improper integral is one in which one or more of the limits of integration is infinite. Such integrals are to be evaluated as regular integrals where the infinity is replaced by a dummy variable, followed by a limit calculation in which the the dummy variable is taken to infinity: \[ \int_a^\infty f(x) \; dx \equiv \lim_{b\to \infty} \int_a^b f(x) \; dx = \lim_{b\to \infty} [ F(b) - F(a) ], \] where $F(x)$ is the anti-derivative function of $f(x)$.
Applications
Later in this chapter, we will learn about the “integral test” for the convergence of a series, which requires the evaluation of an improper integral.