Summary: Let \(X\) be a smooth complete intersection over \(\mathbb{C}\) of dimension \(n-k\) in the projective space \(\mathbf{P}^n_{\mathbb{C}}\), for given positive integers \(n\) and \(k\). For a given integral homology cycle \([\gamma] \in H_{n-k}(X(\mathbb{C}),\mathbb{Z})\), the period integral is defined to be a linear map from the de Rham cohomology group to \(\mathbb{C}\) given by \([\omega] \mapsto \int_\gamma \omega\). The goal of this article is to interpret this period integral as a linear map from the polynomial ring with \(n+k+1\) variables to \(\mathbb{C}\) and use this interpretation to develop a deformation theory of period integrals of \(X\). The period matrix is an invariant defined by the period integrals of the rational de Rham cohomology, which compares the rational structures (\(\mathbb{Q}\)-subspace structures) of the de Rham cohomology over \(\mathbb{C}\) and the singular homology with coefficient \(\mathbb{C}\). As a main result, when \(X'\) is another projective smooth complete intersection variety deformed from \(X\), we provide an explicit formula for the period matrix of \(X'\) in terms of the period matrix of \(X\) and the Bell polynomials evaluated at the deformation data. Our result can be thought of as a modern deformation theoretic treatment of the period integrals based on the Maurer-Cartan equation of a dgla (differential graded Lie algebra).