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:



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_06.gifYou may prefer the trial and error method!