204 Improper Integral
An integral taken as a limit when an endpoint goes to infinity or the function becomes unbounded that is used to assign a finite value to such integrals when the limit exists.
definition [d] (Improper Integral — infinite interval) A definite integral over an infinite interval, defined as a limit:
- \(\displaystyle \int_{a}^{\infty} f(x)\, dx = \lim_{t \to \infty} \int_{a}^{t} f(x)\, dx\) .
- \(\displaystyle \int_{-\infty}^{b} f(x)\, dx = \lim_{t \to -\infty} \int_{t}^{b} f(x)\, dx\) .
If both \(\int_{-\infty}^{a} f(x)\, dx\) and \(\int_{a}^{\infty} f(x)\, dx\) converge, then
- \(\displaystyle \int_{-\infty}^{\infty} f(x)\, dx = \int_{-\infty}^{a} f(x)\, dx + \int_{a}^{\infty} f(x)\, dx\) .
where
- \(f\) is the integrand.
- \(a, b\) are finite endpoints.
- \(x\) is the variable of integration.
- \(t\) is the truncation variable in the limiting process.
Note:
- \(f\) is continuous on the finite part of the interval of integration.
- the improper integral converges if the limit exists as a finite number; otherwise it diverges.
definition [d] (Improper Integral — discontinuous integrand) A definite integral where \(f\) has an infinite discontinuity on \([a,b]\), defined as a one-sided limit of proper integrals:
- (discontinuity at \(b\)) \(\displaystyle \int_{a}^{b} f(x)\, dx = \lim_{t \to b^{-}} \int_{a}^{t} f(x)\, dx\) .
- (discontinuity at \(a\)) \(\displaystyle \int_{a}^{b} f(x)\, dx = \lim_{t \to a^{+}} \int_{t}^{b} f(x)\, dx\) .
If the discontinuity is at an interior point \(c\) with \(a < c < b\), the integral is the sum of the improper integrals from \(a\) to \(c\) and \(c\) to \(b\), provided both converge.
where
- \(f\) is the integrand.
- \(a, b\) are the endpoints of the interval.
- \(c\) is an interior point of discontinuity.
- \(t\) is the truncation variable in the one-sided limit.
Note:
- \(f\) is continuous on each proper subinterval used in the limits.
- the improper integral converges if the limits exist as finite numbers; otherwise it diverges.