The derivative of e^{x} is e^{x}.

This means that ∫ e^{x} dx = e^{x} + 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 ∫ e^{5x} dx the answer could be guessed at e^{5x}. However on differentiating e^{5x} using the Chain Rule we get 5e^{5x} which is 5 times too big so to compensate a factor of 1 / 5 is required.

∫ e

^{5x}dx =

In general:

**Further example**

Find ∫4 e

^{3x+ 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 e^{3x+ 5} dx

Let y = ∫4 e^{3x+ 5} dx

and let u = 3x + 5 so y = ∫4 e^{u} dx

You may prefer the trial and error method!