# Examples of problems that are easier in the infinite case than in the finite case.

I am looking for examples of problems that are easier in the infinite case than in the finite case. I really can’t think of any good ones for now, but I’ll be sure to add some when I do.

One can compute the value of

exactly. This is known as the Gaussian integral, and it has its own Wikipedia page. The answer turns out to be $\frac{1}{2}\sqrt{\pi}.$

But one cannot do the same with

because the antiderivative of the integrand is not an elementary function. This is why we gave a name to the error function which also has its own Wikipedia page.

In that sense, the infinite case is easier than the finite case.

Addendum: The same phenomenon occurs for variants of this integral, in particular we can transform the integrand to evaluate as detailed here on Wikipedia.