Summary: Two classes of functions are hereby considered; namely, \( \eta\)-quasiconvex, and strongly \(\eta\)-quasiconvex functions. For the former, we establish some novel integral inequalities of the trapezoid kind for functions with second derivatives, while, for the latter, we obtain some new estimates of the integral \(\int_{\mathfrak{\alpha }}^{\beta }(\mathfrak{r}-\mathfrak{\alpha })^p(\beta -\mathfrak{r})^q\mathcal{K}(\mathfrak{r}) \,d\mathfrak{r}\) when \(|\mathcal{K}(\mathfrak{r})|\), to some powers, is strongly \(\eta\)-quasiconvex. Results obtained herein contribute to the development of these new classes of functions by providing broader generalizations to some well-known results in the literature. Furthermore, we employ our results to deduce some estimates for the perturbed version of the trapezoidal formula. Finally, applications to some special means are also presented.