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...
If the improper integral converges and $x$ and $y$ are differentiable, we get
$$y'(t)=x'(t)+x(t).$$
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: $$ \frac{d}{dt}\int_{-\infty}^t x(\tau)\,d\tau = 0 + \frac{d}{dt}\int_a^t x(\tau)\,d\tau = x(t). $$
