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A q-integral is a definite integral of a function of q having an expansion in non-negative powers of q for | q | < 1 {|q|<1} (q-series). In his book on hypergeometric series, N. J. Fine [N. J. Fine, Basic Hypergeometric Series and Applications, Math. Surveys Monogr. 27, American Mathematical Society, Providence, 1988] explicitly evaluated three q-integrals. For example, he showed that ∫ 0 e - π ∏ n = 1 ∞ ( 1 - q 2 ⁢ n ) 20 ( 1 - q n ) 16 ⁢ d ⁢ q = 1 16. \int_{0}^{e^{-\pi}}\prod_{n=1}^{\infty}\frac{(1-q^{2n})^{20}}{(1-q^{n})^{16}}% dq=\frac{1}{16}. In this paper, we prove a general theorem which allows us to determine a wide class of integrals of this type. This class includes the three q-integrals evaluated by Fine as well as some of those evaluated by L.-C. Zhang [L.-C. Zhang, Some q-integrals associated with modular forms, J. Math Anal. Appl. 150 1990, 264–273]. It also includes many new evaluations of q-integrals. [ABSTRACT FROM AUTHOR]