A function F (x) is a q-antiderivative of f (x) if D q F (x) = f (x). The q-antiderivative (or q-integral) is denoted by ∫ ∫ f ( x ) d q x and an expression for F (x) can be found from: ∫ ∫ f ( x ) d q x = ( 1 − − q ) ∑ ∑ j = 0 ∞ ∞ x q j f ( x q j ) , which is called the Jackson integral of f (x). For 0 < q < 1, the series converges to a function F (x) on an interval (0, A ] if | f (x) x | is bounded on the interval (0, A ] for some 0 ≤ α < 1.