Let $X$ be a compact complex manifold of complex dimension $n$. The top Chern class of the holomorphic tangent bundle satisfies \begin{align*} \int_{[X]} c_n(TX)=\chi(X), \end{align*} where $[X]$ is the fundamental class determined by the complex orientation, $c_n(TX)$ is represented by any closed real top-degree Chern-Weil representative of the top Chern class, and $\chi(X)$ is the topological Euler characteristic.