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Repeated Integral

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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

int_0^xf(t)dt=F(x)-F(0).
(1)

Now, if F(0)=0, then

F(x)=intf(x)dx=int_0^xf(t)dt.
(2)

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)=int...int_0^x_()_(n)f(x)dx...dx_()_(n)
(3)
=int_0^x(f(t)(x-t)^(n-1))/((n-1)!)dt.
(4)

Similarly, if F(x_0)=F(F(x_0))=...=0, then

int...int_(x_0)^x_()_(n)f(x)dx...dx_()_(n)=int_(x_0)^x(f(t)(x-t)^(n-1))/((n-1)!)dt.
(5)

See also

Fractional Integral, Fubini Theorem, Integral, Multiple Integral

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References

Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach, p. 33, 1993.

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Repeated Integral

Cite this as:

Weisstein, Eric W. "Repeated Integral." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RepeatedIntegral.html

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