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.