We study integrals of the eigenfunctions and associated functions of the formal Sturm--Liouville operator $\Cal L(u)(x)=-\bigl(p(x)\,u^{\prime}(x)\bigr)^{\prime}+q(x)\,u(x)$ defined on a finite interval $G\subset\Bbb R$. We suppose that the complex-valued potential $q=q(x)$ belongs to the class $L_1(G)$ and that piecewise continuously differentiable coefficient $p=p(x)$ has a finite number of the discontinuity points in $G$. Order-sharp upper estimates are established for integrals (over arbitrary closed intervals $[y_1,y_2]\subseteq\overline G$) of the eigenfunctions and associated functions in terms of their $L_2$--norms when $G$ is finite.