Lt ε=0∫ c−εa ƒ(x)dx + Lt ε′=0∫ bc+ε′ ƒ(x)dx,
(1)
where, to fix ideas, b is taken > a, and ε and ε′ are positive. The same definition applies to the case where ƒ(x) becomes infinite, or tends to become infinite, at c, provided both the limits exist. This definition may be otherwise expressed by saying that a partial interval containing the point c is omitted from the interval of integration, and a limit taken by diminishing the breadth of this partial interval indefinitely; in this form it applies to the cases where c is a or b.
Again, when the interval of integration is unlimited to the right, or extends to positively infinite values, we have as a definition
∫ ∞a ƒ(x)dx = Lt h=∞∫ ha ƒ(x)dx,
provided this limit exists. Similar definitions apply to
∫ −∞a ƒ(x)dx, and to ∫ ∞−∞ ƒ(x)dx.
All such definite integrals as the above are said to be “improper.” For example, ∫ ∞0 sin x / x dx is improper in two ways. It means
Lt h=∞ Lt ε=0 ∫ hε sinx/x dx,