Is there a simple formula that would produce the regularized value for the most common divergent integrals?

I know, there is a formula for Cesaro integration, but it is applicable only to Cesaro-summable integrals. Other formulas for regularization essentially convert the integral into a series and regularize that series.

What I am looking for is an integral analog of Faulhaber's formula for Ramanujan's summation:

$$\operatorname{reg} \sum _{n=1}^{\infty} f(n)= -\sum_{n=1}^{\infty} \frac{f^{(n-1)} (0)}{n!} B_n $$

Or even its more universal variant:

$$\operatorname{reg} \sum _{n=1}^{\infty} f(n)=-\frac{f(0)}{2}+i \int_0^{\infty } \frac{f(i t)-f(-i t)}{e^{2 \pi t}-1} \, dt$$

Yes, it is also not universally applicable, but it is universal enough.