  Integration by parts identities (IBPs) can be used to express large numbers
of apparently different d-dimensional Feynman Integrals in terms of a small
subset of so-called master integrals (MIs). Using the IBPs one can moreover
show that the MIs fulfil linear systems of coupled differential equations in
the external invariants. With the increase in number of loops and external
legs, one is left in general with an increasing number of MIs and consequently
also with an increasing number of coupled differential equations, which can
turn out to be very difficult to solve. In this paper we show how studying the
IBPs in fixed integer numbers of dimension d=n with $n \in \mathbb{N}$ one can
extract the information useful to determine a new basis of MIs, whose
differential equations decouple as $d \to n$ and can therefore be more easily
solved as Laurent expansion in (d-n).
