Way to calculate antiderivative if given $n$ values.

Question

If I am given some integral which I don't know how to take, but I know the results of $n$ different definite evaluations, as well as the limits of integration of each, what is the smallest $n$ required to figure out the indefinite expression? Is it even possible? The question could be rephrased as, if given a derivative of a function and the values of $f(a_1)-f(b_1),f(a_2)-f(b_2)...f(a_n)-f(b_n)$, what is the minimum number of values needed to find f.

Answer 1

There is no such $n$.

Finding the indefinite expression is equivalent to being able to determine the value of all definite integrals say in the range $[0,x]$. However, one can always create crazy enough functions that will give you any result over a specific interval, no matter how many existing evaluations you already have.

As an example, consider the function: $$2\alpha^2 xe^{-\alpha^2 x^2}$$ The integral over any interval of the form $[-a,a]$ is zero, and for all intervals $[\epsilon, a]$, where $\epsilon\gg1/\alpha$ the integral is pretty close to zero as well. With this information, you could guess the indefinite expression is pretty close to zero, yet at the point $x=1/\alpha$, the indefinite expression can be as large as you like, depending on your choice of $\alpha$!