Area under Dirac Delta function
When integrating to only \$t\$ there are two cases: if \$t &lt; 0\$ then the integral is \$0\$, if \$t \geq 0\$ then the integral is \$1\$:
$$\int_{-\infty}^{t}\delta(x)dx = \begin{cases} 0\text{, }t &lt; 0 \\ 1\text{, }t \geq 0\end{cases}$$

But this is just another way of writing the unit step function \$u(t)\$ so

$$\int_{-\infty}^{t}\delta(x)dx = u(t)$$

Since $$\lim_{t \to \infty} u(t) = 1$$ then it is also true that

$$\int_{-\infty}^{\infty}\delta(x)dx = 1$$