  Repeated integration is a major topic of multivariable calculus. In this
article, we study repeated integration. In particular, we study repeated
integrals and recurrent integrals. For each of these integrals, we develop
reduction formulae for both the definite as well as indefinite form. These
reduction formulae express these repetitive integrals in terms of single
integrals. We also derive a generalization of the fundamental theorem of
calculus that expresses a definite integral in terms of an indefinite integral
for repeated and recurrent integrals. From the recurrent integral formulae, we
derive some partition identities. Then we provide an explicit formula for the
$n$-th integral of $x^m(\ln x)^{m'}$ in terms of a shifted multiple harmonic
star sum. Additionally, we use this integral to derive new expressions for the
harmonic sum and repeated harmonic sum.
