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The theory of the definite integral developed from the Riemann sum requires that the integrand be a bounded function defined on a finite domain. If the domain is unbounded, or if the function itself is not bounded, then such a definite integral is called improper, and its value is given by the limiting processes defined in Table 4.5.1.
Type

Characteristic

Definition

1.

Unbounded domain
($c$ is any point of continuity of $f$)

${\int}_{a}^{\infty}f\left(x\right)\mathit{DifferentialD;}x\equiv \underset{t\to \infty}{lim}{\int}_{a}^{t}f\left(x\right)\mathit{DifferentialD;}x$
${\int}_{\infty}^{a}f\left(x\right)\mathit{DifferentialD;}x\equiv \underset{t\to \infty}{lim}{\int}_{t}^{a}f\left(x\right)\mathit{DifferentialD;}x$
${\int}_{\infty}^{\infty}f\left(x\right)\mathit{DifferentialD;}x\equiv \underset{s\to \infty}{lim}{\int}_{s}^{c}f\left(x\right)\mathit{DifferentialD;}xplus;\underset{t\to \infty}{lim}{\int}_{c}^{t}f\left(x\right)\mathit{DifferentialD;}x$

2a.

Unbounded integrand
Asymptote at an endpoint

${\int}_{a}^{b}f\left(x\right)\mathit{DifferentialD;}x$ ≡ $\{\begin{array}{cc}\underset{t\to a\+}{lim}{\int}_{t}^{b}f\left(x\right)\mathit{DifferentialD;}x& \mathrm{asymptote}\mathrm{at}xequals;a\\ \underset{t\to b}{lim}{\int}_{a}^{t}f\left(x\right)\mathit{DifferentialD;}x& \mathrm{asymptote}\mathrm{at}xequals;b\end{array}$

2b.

Unbounded integrand
Asymptote at $c$, an interior point

${\int}_{a}^{b}f\left(x\right)\mathit{DifferentialD;}xequals;\underset{s\to \mathrm{c}}{lim}{\int}_{a}^{s}f\left(x\right)\mathit{DifferentialD;}xplus;\underset{t\to cplus;}{lim}{\int}_{t}^{b}f\left(x\right)\mathit{DifferentialD;}x$

Table 4.5.1 Definitions of three different types of improper Riemann integrals



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An improper integral that has a finite value is said to converge; otherwise it is said to diverge.
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Where there is a sum of two limits (cases 1 and 2b), the limits must be taken separately. Setting $s\=t$, and taking the limits simultaneously results in the Cauchy Principal Value (CPV) of the integral, which can exist while the value defined by Table 4.5.1 might not. On the other hand, if the value as per Table 4.5.1 exists, then it will equal the CPV.
Theorem 4.5.1 can be used to determine the convergence or divergence of an improper integral for which an antiderivative is either difficult or impossible to determine.
Theorem 4.5.1: Comparison Test for Improper Integrals

1.

$f$ and $g$ are continuous functions

2.

$f\left(x\right)\ge g\left(x\right)\ge 0$

⇒
1.

${\int}_{a}^{\infty}f\left(x\right)\mathit{DifferentialD;}x$ converges ⇒${\int}_{a}^{\infty}g\left(x\right)\mathit{DifferentialD;}x$ converges

2.

${\int}_{a}^{\infty}g\left(x\right)\mathit{DifferentialD;}x$ diverges⇒${\int}_{a}^{\infty}f\left(x\right)\mathit{DifferentialD;}x$ diverges




In words, Theorem 4.5.1 might be expressed as follows. The convergence of the larger function forces the convergence of the smaller; the divergence of the smaller function forces the divergence of the larger.