A Simple Integration Question Involving Square Roots

Question

1. Let's say I have a function $f(x)$ of which I have its indefinite integral, but not the function itself. How would I find the integral of $\sqrt{f(x)}$?
2. If it is not possible, then let's say I do know what $f(x)$ is? How would I then find the integral $\sqrt{f(x)}$, knowing $f(x)$?

Answer 1

Example: $$f(x) = x + \cos x.$$ The indefinite integral of $f(x)$ is trivial to find, but that doesn't help us find the indefinite integral of $\sqrt{f(x)}$—indeed that indefinite integral is not an elementary function.

Not all integrals can be evaluated!