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  is the integral of , then

(1)

Now, if , then

(2)

It follows by induction that if , then the -fold integral of  is given by

			(3)
			(4)

Similarly, if , then

(5)

REFERENCES:

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