Let $X$ be a complex variety. By Poincare's lemma, its singular cohomology can be computed as hypercohomology of the holomorphic de Rham complex (viewing $X$ as a complex manifold)
$$\text{H}^\cdot(X,\mathbf{C})\ \simeq\ \text{H}^\cdot(X,\Omega^\bullet).$$
By GAGA the right side is the same as the hypercohomology of the complex of algebraic de Rham complex.
In any case, you can compute the hypercohomology $ \text{H}^\cdot(X,\Omega^\bullet)$ using Cech cohomology, in terms of an open cover $\{U_\alpha\}$ of $X$. For instance, if $X$ has dimension $n$, a representative of $\text{H}^n(X,\Omega^\bullet)$ is a collection of $n$ forms
$$\omega_{n,\alpha}\ \in\ \text{H}^0(U_\alpha,\Omega^n)$$
which do not quite glue to a global $n$-form, but there are $n-1$ forms
$$\omega_{n-1,\alpha,\beta}\ \in\ \text{H}^0(U_\alpha\cap U_\beta,\Omega^{n-1})$$
such that $d\omega_{n-1,\alpha,\beta}=\omega_\alpha-\omega_\beta$ on the intersection. Similarly, there are $n-2$ forms $\omega_{n-2,\alpha,\beta,\gamma}$ such that $d\omega_{n-2,\alpha,\beta,\gamma}=\omega_{n-1,\alpha,\beta}-\omega_{n-1,\alpha,\gamma}+\omega_{n-1,\beta,\gamma}$, and so on.
For example, on $X=\mathbf{P}^1$ with its standard cover by two affine lines, $\text{H}^2(\mathbf{P}^1)$ is generated by the meromorphic differential form
$$dx/x\ \in\ \text{H}^0(\mathbf{A}^1_0\cap\mathbf{A}^1_\infty, \Omega^1). $$
Question: Assume that $X$ is smooth and proper. We then know that an element of $\text{H}^\cdot(X,\mathbf{C}) \simeq \text{H}^\cdot(X,\Omega^\bullet)$ can be integrated. What does integration look like in terms of these hypercohomology classes?
Looking at the $\mathbf{P}^1$ example, one might guess that the answer has something interesting to do with residues.
Best Answer
One way to see this is algebraically. Let $\omega_X = \Omega^n_X$. The map you seek to describe is $$ \int_X \colon \mathrm{H}^n(X,\omega_X) \to \mathbb{C} $$ Abstractly, it arises as the counit of the Serre duality adjunction $$ \mathrm{H}^n(X,-) \dashv -\otimes \omega_X $$ By the compatibility between local and global duality, there is a commutative diagram $\require{AMScd}$ \begin{CD} \otimes_P \mathrm{H}^n_P(\omega_X)@>can>> \mathrm{H}^n(X,\omega_X)\\ @V \otimes_P \mathrm{res}_P V V @VV\int_XV\\ \mathbb{C} @= \mathbb{C} \end{CD} and the maps $\mathrm{res}_P$ that arise by local duality, are computed through higher dimensional residues. This is explained with great detail in J. Lipman's blue book:
Dualizing sheaves, differentials and residues on algebraic varieties. Astérisque, No. 117 (1984).
For the comparison between the algebraic and the analytic case and the $\frac{1}{(2 \pi i)^n}$ factor (up to a sign), see
Sastry & Tong, The Grothendieck Trace and the de Rham Integral, Canad. Math. Bull. Vol. 46 (3), 2003 pp. 429–440.
To make things more explicit, you may assume that your covering $\{U_i\}_{i \in I}$ is finite and each $U_i$ is the complement of a certain section $f_i$ of a line bundle. In this case a Czech cocycle is defined by a residual symbol $$ \left[ \frac{ w } { f_0 \dots f_n } \right] $$ with $w$ a meromorphic $n$-form. This is described on page 195 of
Hartshorne, Residues and duality. Lecture Notes in Mathematics, 20, Springer-Verlag, 1966.
So the objective is to compute $$ \int_X \left[ \frac{ w } { f_0 \dots f_n } \right] $$
The Czech-cocycle/symbol induces a certain element
$$ \left[ \frac{ w } { f_0 \dots \hat{f_i} \dots f_n } \right]_P $$ in $\mathrm{H}^n_P(\omega_X)$ where $p\in U_i$ so $(f_i)_P$ is invertible in $\mathcal{O}_{X,P}$. This element is describable using a Koszul complex to compute $\mathrm{H}^n_P(\omega_X)$. Finally the computation of the integral is reduced to compute $$ \mathrm{res}_P \left[ \frac{ w } { f_0 \dots \hat{f_i} \dots f_n } \right]_P $$. This can be done explicitly, over the points $P$ that are poles of the differential form $w$. Locally at $P$ $$ w = \phi f_0 \wedge \hat{f_i} \wedge f_n $$ with $\phi$ in the fraction field of $\mathcal{O}_{X,P}$. Then, $$ \mathrm{res}_P \left[ \frac{ w } { f_0 \dots \hat{f_i} \dots f_n } \right]_P = a_{(-1, \dots, -1)} $$ where $a_{(-1, \dots, -1)} \in \mathbb{C}$ denotes the corresponding coefficient in the Laurent expansion of $\phi$. This is described using the identification of $\widehat{\mathcal{O}}_{X,P}$ with the power series ring $\mathbb{C}[[f_0 \dots \hat{f_i} \dots f_n]]$.