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)

\begin{align} \int e^x \: dx = \frac{e^x}{\ln e} + C = e^x + C \end{align}

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)

\begin{align} \int 5^x \: dx \\ = \frac{5^x}{\ln 5} + C \end{align}

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

(3)

\begin{align} \quad \frac{d}{dx} ( \frac{5^x}{\ln 5} + C ) = \frac{1}{\ln 5} \cdot \frac{d}{dx} ( 5^x + C ) \\ \frac{d}{dx} ( \frac{5^x}{\ln 5} + C ) = \frac{1}{\ln 5} \cdot 5^x \ln 5 \\ \frac{d}{dx} ( \frac{5^x}{\ln 5} + C ) = 5^x \end{align}

Example 2

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

We will once again apply what we have learned:

(4)

\begin{align} \int 3^x + 6^x \: dx = \int 3^x \: dx + \int 6^x \: dx \\ \int 3^x + 6^x \: dx = \frac{3^x}{\ln 3} + \frac{6^x}{\ln 6} + C \end{align}

Example 3

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

Applying what we have learned, we get that:

(5)

\begin{align} \int 4x + 4^x \: dx = 4\frac{x^2}{2} + \frac{4^x}{\ln 4} + C \\ \int 4x + 4^x \: dx = 2x^2 + \frac{4^x}{\ln 4} + C \end{align}