Simpson's rule for improper integrals

Question

Let $$F(x)=\int_{-\infty}^x f(t)dt,$$ where $x\in\mathcal{R}$, $f\geq 0$ is complicated (it cannot be integrated analytically).

Can I used the Simpson's rule to approximate this integral, knowing that $f(-\infty)=0$?

Answer 1

Yes. You can do a change of variable in your integral $\xi = x-\frac{1-t}{t}$, $d\xi = \frac{1}{t^2}dt$ and integrate for $0<\xi<1$. There are other methods too. See Wikipedia.

Answer 2

Using estimates on $f$, you can obtain some $M>0$ such that $\int_{-\infty}^{-M}f(t) dt < \frac{\varepsilon}{2}$ and then use Simpson's rule with enough points so that you approximate $\int_{-M}^x f(t) dt$ with an error smaller than $\frac{\varepsilon}{2}$. This guarantees an overall error smaller than $\varepsilon$.