Recently I have encountering the dirac delta function $\delta(x)$ more often. And have been thinking about its antiderivatives:
The function is defined as: $$\delta(x)=\begin{cases} \infty ; x=0\\ 0 ; \text{elsewhere} \end{cases}$$
Its antiderivative can be defined as: The function is defined as: $$\int_{-\infty}^t \delta(x)dx=\begin{cases} 0 ; t <0\\ 1 ; t\geq0 \end{cases}$$
This is the definition of the unit step function. Hence one can claim: The function is defined as: $$\int_{-\infty}^t \delta(x)dx=u(t)$$ But now suppose one took an indefinite integral of the function, would the antiderivative look like: $$\int \delta(x)dx=u(t) +c$$ Or if one went back to the definition: The function is defined as: $$\int_{t_0}^t \delta(x)dx=\begin{cases} 1 ; t\cdot t_0 \leq 0\\ 0 ; t_0\cdot t>0 \end{cases}=c\cdot u(t)$$ where $c$ is the constant of integration. Where do I put the constant of integration?