I know that in general, there exist more functions which are integrable than there are functions which are differentiable (nowhere differentiable to be exact), at least in $C([0,1])$, by the Baire Category Theorem. However, in general most elementary functions we know tend to have a derivative, which is generally almost always easier to compute than the integral. I have read in many places that functions or simulations where we need to compute either the derivative or integral analytically, the integral is easier. I can't find or think of any examples. Does anyone have any to shed light into this claim? Thanks!
Answer
I do not know, if this counts as an example, but let $f$ denote any elementary, but "complicated" function, for example
$$ f(x) = \exp\bigl(\sin x + \sqrt[23]{\cos x + 34}\bigr) + \sqrt[x]{x + \sin x} $$
or whatever. Then define $g(x) := \frac{f'(\log x)}x$. Then $f$ is an elementary function, the antiderivative of $g$ is $f \circ \log$ (quite easy), and although it is possible to compute $g'$ analytically (first of course one has to compute $f'$), I will not do this here.
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