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Integration of the power flowing through an element

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 :

equation_1

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

equation_2

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

equation_3

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

(1)equation_4

Substituting the function in the integral yields :

(2)equation_5

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

Substituting into the equation (2) :

equation_7

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

equation_8