What is the meaning of positive and negative shift in signal correlation?
Your Eq(3), for positive shift, is correct. And your Eq(5), for negative shift, should also be the same, but is wrong for $\tau &lt; 0$ case. 

The lower limit of the integral in Eq(4) should be $-\tau$ but you took it $\tau$. The reason is the step function $u(t+\tau)$ will have the nonzero range for $t &gt; -\tau$.

You will get the same result with the correct limit.