at some point in graduate school, everyone encounters the rather abstract theorem that every compact topological group possess a unique translation-invariant probability measure, known as the haar measure. rarely, if ever, is the concrete problem of computing integrals under this measure addressed. i will explain an analogue of the fundamental theorem of calculus in this setting: a theorem which reduces the computation of a large class of group integrals to a symbolic problem. this symbolic problem can, in some important cases, be cleverly solved using techniques from algebraic combinatorics.