Integral over operators equal -When are operators equal?

Question

This is a physics motivated question from quantum mechanics. When I have two operators $\hat{A}$ and $\hat{B}$ acting on the Hilbertspace $\mathbb{C}-L^2(\mathbb{R})$ with $$\int\psi^*(x)\hat{A}\psi(x)\:\mathrm{d}x= \int\psi^*(x)\hat{B}\psi(x)\:\mathrm{d}x $$ and where the initial conditions of $\psi$ can be arbitrary how can I then conclude $\hat{A}= \hat{B}$?