Is it possible to replace an integrator system with an equivalent differentiator?

Question

I have a block diagram which has the input-output relation as follows:

$y(t)=x(t)+\int_{-\infty }^{t} x(\tau) d\tau$

Can I create the equivalent system by using differentiators rather than integrators? I think something like taking derivative of both sides of the equation but improper integral makes it hard. Is there any nice way to convert this system? Thanks in advance...

Answer 1

About your doubts with the improper integral: supposing convergence (as Fred as pointed in the other answer) and taking any constant $a\in\Bbb R$, $$ \int_{-\infty}^t x(\tau)\,d\tau = \int_{-\infty}^a x(\tau)\,d\tau + \int_a^t x(\tau)\,d\tau. $$ The integral from $-\infty$ to $a$ is constant and applying the FTC: $$ \frac{d}{dt}\int_{-\infty}^t x(\tau)\,d\tau = 0 + \frac{d}{dt}\int_a^t x(\tau)\,d\tau = x(t). $$

Answer 2

If the improper integral converges and $x$ and $y$ are differentiable, we get

$$y'(t)=x'(t)+x(t).$$