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.