I need a hint on how to attack the following integral. $$\int_0^\infty \mathrm{e}^{-f(x)} \dfrac{\mathrm{d}^3 f(x)}{\mathrm{d}x^3} \,\mathrm{d}x.$$ I tried it by part and substitution, but I did not succeed. I know that the integral exists and that I can interchange derivative and integral and that $$\lim_{x\mapsto\infty} f(x) = 0.$$ Moreover $f(0)$ exists and $f \in C^{\infty}$.
Thanks a lot!
My attempt: I tried to integrate by part twice where the function to derive is always $e^{-f(x)}$ and integrate the rest. My idea was to arrive at something similar to $e^{-f(x)}\, f(x)$ and then solve it easily by substitution. Unluckily, I arrived to evaluate $$ \int_0^{\infty} e^{-f(x)} \left(\dfrac{\mathrm{d}f(x)}{\mathrm{d}x}\right)^3 \,\mathrm{d}x.$$ which still I'm not able to solve.