Let \((M,g)\) be a compact orientable Riemannian manifold, \(\dim M =n =2m\), \(\nabla\) the Riemannian connection, \(\omega\) the connection form and \(\Omega =d\omega- \omega\wedge \omega\) the curvature form. The structure group of the tangent bundle of \(M\) is \(SO(n)\), and the corresponding Lie algebra \(o(n)\) is the algebra of real \((n\times n)\)-skew symmetric matrices. Let \(F\) be an \(SO(n)\)-invariant polynomial of degree \(k\) defined on \(o(n)\). Then \(F(\Omega)\) is the characteristic form of \(M\) defined by \(F\). In this paper, the author calculates the integral \(\int_M F(\Omega)\wedge \sigma\) when \(\text{deg}(F)=k< m\), \(\sigma\) being an arbitrary closed \((n-2k)\)-form of \(M\). This problem was solved by \textit{R. Bott} [J. Differ. Geom. 1, 311-330 (1967; Zbl 0179.28801)] and by \textit{P. F. Baum} and \textit{J. Cheeger} [Topology 8, 173-193 (1969; Zbl 0179.28802)] when \(\text{deg}(F)=m\). All known results about this problem are contained, as particular cases, in the result of this paper.