Find the autocorrelation function of signal $x(t) = u(t) - u(t-1)$
Don't make this more complicated than it really is. $x(t)$ is non-zero in the interval $t\in[0,1]$, and $x(t+\tau)$ is non-zero in the interval $t\in[-\tau,1-\tau]$. The integrand is non-zero only if the two functions overlap. There is no overlap for $1-\tau&lt;0$ and $-\tau&gt;1$, i.e., for $|\tau|&gt;1$. So for $|\tau|&gt;1$ the autocorrelation is zero.
For $-1&lt;\tau&lt;1$, we have, according to above considerations, the following integral:
$$\mathcal{R}_{xx}(\tau)=\int_{\max\{0,-\tau\}}^{\min\{1,1-\tau\}}dt=\begin{cases}\displaystyle\int_{0}^{1-\tau}dt,&amp;0&lt;\tau&lt;1\\\displaystyle\int_{-\tau}^{1}dt,&amp;-1&lt;\tau&lt;0\end{cases}$$