Let $X$ be a $n$-dimensional compact connected complex manifold and $V$ its connected submanifold of dimension $k$. Let $D_1,\cdots, D_k$ be divisors on $X$.
Each of the line bundles $\mathcal{O}_X\left(D_i\right)$ has a first Chern Chern class
$$ c_1\left(\mathcal{O}_X\left(D_i\right)\right) \in H^2(X ; \mathbf{Z}), $$ The cup product of these classes is then an element $$ c_1\left(\mathcal{O}_X\left(D_1\right)\right) \cdot \ldots \cdot c_1\left(\mathcal{O}_X\left(D_k\right)\right) \in H^{2 k}(X ; \mathbf{Z}): $$ Denoting by $[V] \in H_{2 k}(X ; \mathbf{Z})$ the fundamental class of $V$, cap product leads finally to an integer $$ \left(c_1\left(\mathcal{O}_X\left(D_1\right)\right) \cdot \ldots \cdot c_1\left(\mathcal{O}_X\left(D_k\right)\right)\right) \cap[V] \in H_0(X ; \mathbf{Z})=\mathbf{Z} $$ $\left(c_1\left(\mathcal{O}_X\left(D_1\right)\right) \cdot \ldots \cdot c_1\left(\mathcal{O}_X\left(D_k\right)\right)\right) \cap[V]$ is called the intersection number.
In many occasions, I have heard many individuals assert that the intersection number is equal to the integral $\int_V c_1\left(\mathcal{O}_X\left(D_1\right)\right) \wedge \ldots \wedge c_1\left(\mathcal{O}_X\left(D_k\right)\right)$. (Here we use the notation $c_1\left(\mathcal{O}_X\left(D_i\right)\right)$ to denote any differential form that serves as a representative of this class). I also think this formula is correct and I think many references have used this formula implicitly.
Question: 1, How to rigorously prove this formula: $$\left(c_1\left(\mathcal{O}_X\left(D_1\right)\right) \cdot \ldots > \cdot c_1\left(\mathcal{O}_X\left(D_k\right)\right)\right) \cap[V]= > \int_V c_1\left(\mathcal{O}_X\left(D_1\right)\right) \wedge \ldots > \wedge c_1\left(\mathcal{O}_X\left(D_k\right)\right) ?$$
2, If we replace the submanifold $V$ by a singular subvariety, does this formula also hold?
