This means that ∫ ex dx = ex + c
In other words the basic exponential function differentiates and integrates to itself.
There are two methods for integrating more complex exponential functions.
Integrating exponential functions by inspection
This is in effect a trial and error method, but becomes quite easy with practice.
To find ∫ e5x dx the answer could be guessed at e5x. However on differentiating e5x using the Chain Rule we get 5e5x which is 5 times too big so to compensate a factor of 1 / 5 is required.
∫ e5x dx =
Find ∫4 e3x+ 5 dx =
Integrating exponential functions by substitution
Some people prefer a longer but more methodical method and the following substitution can be used.
Find ∫4 e3x+ 5 dx
Let y = ∫4 e3x+ 5 dx
and let u = 3x + 5 so y = ∫4 eu dx
You may prefer the trial and error method!