Where is the error? Application of FTC

Question

Please, I need a help to see the error on this argument:$$\int_0^tf(a)g(t-a)da=\int_0^tf(t-a)g(a)da\implies$$

$$\dfrac{d}{dt}\int_0^tf(a)g(t-a)da=\dfrac{d}{dt}\int_0^tf(t-a)g(a)da\implies$$

$$f(t)g(0)=f(0)g(t)$$

If $f(0)=g(0)=k\neq 0$, então

$$f(t)=g(t)\forall t\in\Bbb R.$$

It is obviously incorrect, but I could not find the error.

Many thanks!

Answer 1

You can't apply FTC here -- you have to use the Leibniz integral rule. This is because not only are the bounds changing, but the inside of the integral itself is changing as well. Contrast this with $$\frac{d}{dx}\int_0^xxt\;dt=\frac{t^2}2,$$ not $xt$ as a naive application of FTC would entail.