Integration of Exponential Functions

VectorMen.jpgThe derivative of ex is ex.

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 = Y12_Integration_of_Exponential_Functions_01.gif

In general:

Y12_Integration_of_Exponential_Functions_02.gif
Y12_Integration_of_Exponential_Functions_03.gif

 

Further example

Find ∫4 e3x+ 5 dx = Y12_Integration_of_Exponential_Functions_04.gif

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

Y12_Integration_of_Exponential_Functions_05.gif

Y12_Integration_of_Exponential_Functions_06.gifYou may prefer the trial and error method!