In calculus, an improper integral is defined as the limit of a definite integral, as an endpoint, or both endpoints, of the interval approaches either a specified real number or ∞ or −∞. The integral
Such an integral is often written symbolically just like a standard definite integral, perhaps with infinity as a limit of integration. But that conceals the limiting process. By using the more advanced Lebesgue integral, rather than the Riemann integral, one can in some cases bypass this requirement, but if one simply wants to evaluate the limit to a definite answer, that technical fix may not necessarily help. It is more or less essential in the theoretical treatment for the Fourier transform, with pervasive use of integrals over the whole real line.
Table of contents |
2 Vertical asymptotes at bounds of integration 3 Cauchy principal values |
The most basic of improper integrals are integrals such as: ∫0∞ dx / (x2 + 1). Such an integral can be evaluted by noting the antiderivative: arctan x. The integral is
Consider ∫01 dx / x(2/3). This integral involves a function with a vertical asymptote at x = 0.
One can evaluate this integral by evaluating from b to 1, and then take the limit as b approaches 0. One should note that the antiderivative of the above function is (3/2)(x(2/3)); which can be evaluated by direct substition: (3/2)(1 − b(2/3)). The limit as b approaches 0 equals: (3/2) − 0 = 3/2 = 1.5.Infinite bounds of integration
Vertical asymptotes at bounds of integration