Repeated Integral
A repeated integral is an integral taken multiple times over a single variable (as distinguished from a multiple integral, which consists of a number of integrals taken with respect to different variables). The first fundamental theorem of calculus states that if F(x)=D^(-1)f(x) is the integral of f(x), then
| [画像: int_0^xf(t)dt=F(x)-F(0). ] |
(1)
|
Now, if F(0)=0, then
It follows by induction that if F(0)=F(F(0))=...=0, then the n-fold integral of f(x) is given by
D^(-n)f(x) = [画像:int...int_0^x_()_(n)f(x)dx...dx_()_(n)]
(3)
Similarly, if F(x_0)=F(F(x_0))=...=0, then
See also
Fractional Integral, Fubini Theorem, Integral, Multiple IntegralExplore with Wolfram|Alpha
WolframAlpha
More things to try:
References
Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach, p. 33, 1993.Referenced on Wolfram|Alpha
Repeated IntegralCite this as:
Weisstein, Eric W. "Repeated Integral." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RepeatedIntegral.html