This is probably a very stupid question but I came across the following in an electrical engineering textbook
Here is what I did:
I am wondering, in Eq 6.5, why have function of current changed from being a vairbale of t to a variable of tau? And also in Eq 6.6, where is the v(t0) coming from?
Again, I apologise if this is really basic, but I haven't done integration problems in years and I couldn't find anything talking specifically about variable changes.
$2.$ The constant of integration $\displaystyle v(t_0)=\frac1C\int_{-\infty}^0i\;$ is the voltage that the capacitor had started with at arbitrary time $t_0.$
$1.$ Notice that my integral in point 2 above has neither $\mathrm dt$ nor $\mathrm d\tau?$ That's because they are just dummy variables (here understood to be representing time), and dropping the integration variable here in favour of a succinct notation loses no generality.$\quad$In Eqn 6.5, the author had switched the integration variable from $\color{#00F}t$ to $\color{#00F}\tau$ (any dummy variable will do, since it doesn't explicitly appear in the integrand) in order to distinguish the integration variable from the upper limit of integration $\color{#C00}t$ (i.e., the $\color{#C00}t$ in the interval of integration $[-\infty,\color{#C00}t]$).
(Check out the statement of the Fundamental Theorem of Calculus Part 1; similarly as above, because $\color{#C00}x$ is already the upper limit of integration, $\color{#00F}t$ is instead chosen as the variable of integration.)