# Indefinite Integrals of Exponential Functions

Theorem 2: If $f(x) = a^x$ then $\int a^x \: dx = \frac{a^x}{\ln a} + C$. Furthermore, if $f(x) = e^x$, then $\int e^x \: dx = e^x + C$. |

The property above is rather easy to visualize with the function $f(x) = e^x$:

(1)Since $\ln e = 1$, we get that the integral of $e^x$ is simply $e^x$, which should make sense since the derivative of $e^x$ is also $e^x$.

Let's now look at some examples.

## Example 1

**Evaluate the following integral: $\int 5^x \: dx$, and then verify that it is in fact the antiderivative by differentiating the result.**

By applying the property above, we can easily simplify this:

(2)Now let's verify our result by differentiating the function. We should get $5^x$ if we have done everything correctly.

(3)## Example 2

**Evaluate the following integral $\int 3^x + 6^x \: dx$.**

We will once again apply what we have learned:

(4)## Example 3

**Evaluate the following integral $\int 4x + 4^x \: dx$.**

Applying what we have learned, we get that:

(5)