I am currently learning about electric circuits and came across some math procedures that I have not used in a long time. Such as integrating a exponential function of the eulers number.

Take an electric element that is subjected to a current and therefore a voltage between its terminals.

We know that :

- the voltage across its terminals is a function of time, v=2e
^{-t} - the current flowing through it is also a function of time, i=3e
^{-t}

We want to know the amount of electrical charge goes into to element in a time period from 0 to 10 seconds. We know that power is the time rate of change of the charge :

Then the charge can be found by integrating the power in a time period:

Substituting the power by its constituents, voltage and current we get :

To integrate the function of t define u as a function of t :

Substituting the function in the integral yields :

Then, by derivating both sides of the equation (1) we get :

Substituting into the equation (2) :

Then, to calculate the amount of charge between instants 0 and 10 just :