Let $\mathbb{P}^n$ denote the complex $n$-dimensional projective space, also let $\mathcal{O}(1)$ denote the hyperplane bundle, i.e. the dual bundle of the tautological bundle $\mathcal{O}(-1)$.
I have seen that $$ \int_{\mathbb{P^n}} \mathrm{c}^n_1(\mathcal{O}(1)) = 1\,, $$ where $\mathrm{c}_1$ denotes the first Chern class. A reference for this fact may be found here at section 3.6.1 (pdf page 75). There is given a justification that it is not understandable by me. I am aware of a computational proof in the case of $\mathbb{P}^1$. The proof follows by constructing a hermitian structure on $\mathcal{O}(1)$ and then the Chern connection of this hermitian structure. The curvature form of this connection equals the first Chern form (multipled by the constant $\frac{i}{2\pi}$), and its integral is computed easily to $1$.
In the general case the same approach gives some really nasty calculations, that a have not gone through fully. To be precise, the first Chern form is given, locally on $U_0 = \{[1: w_1 : \cdots : w_n] \,|\, w \in \mathbb{C}\} \subset \mathbb{P}^n$, by $$ x = \frac{i}{2\pi} \frac{1}{(1 + |w|^2)^2}\sum_{ij}\bigl(\delta_{ij}(1 + |w|^2)^2 - \bar w_iw_j\bigr) dw_i \wedge d\bar w_j\,. $$ I have not calculated the form $x^n$ neither the integral $\int_{\mathbb{P}^n} x^n$.
I would like to now if there is a way to prove the claimed equality in the general case through such a computational approach. Further, I would like to be provided any reference with a proof of this claim, since the only one that I have found is the one I linked and this one was not very helpful for me. Let me state that I have studied all the above following the differential geometry way. That means I know the definition of characteristic classes that it uses the curvature form and that I have minimal experience with algebraic geometry. Thank you.