and use the integrating variable \(z\ =\ t-t_{in}\ \) , so that,   \(t_{in}<t\ \le t_{in}+\Delta t\) becomes \(0\ <\ T\ \le\Delta t\) ,   \(dz=-dx\)  and the extremes of integration move such \(x=t_{in\ }\ \to z=t-t_{in}\ =T\) and \(x=t\ \to z=0\)  :