TITLE: What is the relationship between $\int f(x) \, dx$ and $\int 1/f(x) \, dx$?
QUESTION [5 upvotes]: So many integration problems would be made so easy if we were just allowed to work with the reciprocal of a function. Is there a way to relate $\int f(x) \, dx$ and $\int 1/f(x) \, dx$ ? How about if you had limits of integration? Thanks.
P.S. If there is no relationship, is it proven that there is no relationship or has someone just not found it yet?

REPLY [3 votes]: Note that for $x\gt0$, the minimum of $x+\frac1x$ is $2$; that is,
$$
x+\frac1x-2=\left(\sqrt{x}-\frac1{\sqrt{x}}\right)^2\ge0
$$
Therefore, we get for $f\gt0$,
$$
\int_a^b\frac1{f(x)}\mathrm{d}x+\int_a^bf(x)\,\mathrm{d}x\ge2(b-a)
$$