The Jackson integral represents a q-analogue of the Riemann integral, thereby extending the integration concept into the domain of q-calculus, while the Riemann integral remains a traditional calculus tool for assessing the area under a curve. It is well known that Henstock-Kurzweil integral is a generalized Riemann integral. In this article, we introduce q-analogue of Henstock-Kurzweil integral, called q-Henstock-Kurzweil integral. We discuss several important properties of newly introduce q-Henstock-Kurzweil integrals and its some results. Moreover, we show that q-Henstock-Kurzweil integrable functions contain Henstock-Kurzweil integrable functions. Furthermore, we introduce Fundamental Theorem of Calculus for q-Henstock-Kurzweil integrable functions in q-analogous approach. Finally, using this integrable functions we suggest a solution method for a class of linear fractional q-differential equations.
Zengin et al. (2025) studied this question.