Monotonic operations and integrals

Question

If I have a monotonic function, say ln, can I bring it inside an integral?

in other words, is $$\ln\left[ \int f(x)\, dx\right] = \int \ln(f(x))\, dx.$$

My limits of integration don't depend on $x,$ so I think I can.

Could I move expectation inside the integral?

(I am working on something with the Cramer-Rao inequality, which requires that I take the natural log and then the expectation of a function.)

Answer 1

$\ln\left(\int_0^1 x\,dx\right)=\ln\left(\frac{1}{2}\right),$ but $\int_0^1 \ln x\,dx=-1$. So no, it's not that simple.

Answer 2

No. As a counterexample, take $f(x)=e^x$.