Apostol asks us to check if the equation of
$$y_1 = \int\limits_{c}^x f(x - t)R(t)dt$$
is a particular solution of the differential equation $y'' + ay' + by = R$, s.t. $f(0) = 0, f'(0) = 1$.
The correct solution involves the application of Leibniz rule to $y_1$ to find its derivative. BUT why doesn't the first fundamental theorem of calculus work? Then,
$$\frac{dy_1}{dx} = \frac{d}{dx}\int\limits_{c}^x f(x - t)R(t)dt = f(x - x)R(x) = 0!$$
which is incorrect. What went wrong here?