Fundamental Theorem of q-Calculus and Integration by Parts

In ordinary calculus, a derivative is defined as the limit of a ratio, and a definite integral is defined as the limit of an infinite sum. Their subtle and surprising relation is given by the Newton-Leibniz formula, also called the fundamental theorem of calculus. In contrast, since the introduction of the definite q-integral has been motivated by an antiderivative, the relation between the q-derivative and definite q-integral is more obvious. Analogous to the ordinary case, we have the following fundamental theorem, or Newton-Leibniz formula, for q-calculus.