When we have a relationship like the following
$$V=L\frac{di}{dt}$$
then in order to find the current $i$ we have to integrate.
My question is what how we choose what kind of integration we must apply. In some textbooks they integrate this very relationship like this:
$$i=\frac{1}{L}\int Vdt+c $$
and in some like this:
$$i=\frac{1}{L}\int_{t_{0}}^{t} Vd\tau+c $$
How do I choose if I have to integrate with indefinite or definite integral?
Let $F(t)$ an antiderivative of $V(t)$. The second integral is: $$ \int_{t_0}^t V d \tau= F(t)-F(t_0) $$ where $F(t_0)=k$ is a constant tha is the value of $F$ at the time $t_0$. So, write $$ \int_{t_0}^t V d \tau+c= F(t)-F(t_0)+c $$ can have a sense only if you know that $c$ is the value of $F$ at some time different from $t_0$.
Anyway, The difference from $$ \int_{t_0}^t V d\tau $$ and $$ \int V dt+c $$ is that in the first the value of constant of integreation is specified by the limit of integration $t_0$, in the second the value of the constat have to be specified by some added initial condition.
