I am interested in representing the improper integral of a function $f(x)$ as a series the following way:
$$p_k=\frac{\int_k^{k+1} x f(x) \, ds}{\int_k^{k+1} f(x) \, dx}$$
$$\int_0^\infty f(x)=\int_0^{p(0)} f(x) \, dx+\sum _{k=1}^{\infty } \int_{p(k-1)}^{p(k)} f(x) \, dx$$
In case the sum does not converge, a regularization technique (Ramanujan, Zeta, Dirichlet) should be used.
What would be the values of such integrals for functions $1/(x+1)$, $x$, $x^2$?