In this paper, we consider the class of $p$-convex functions. We derive some new integral inequalities of Hermite--Hadamard and Simpson type for differentiable $p$-convex functions using two new integral identities. Some special cases are also discussed. Interested readers may find novel and innovative applications of $p$-convex functions in various branches of pure and applied sciences. The ideas and techniques of this paper may stimulate further research in this field.