Calculus Examples

$e^{a x}$

Step 1

Write $e^{a x}$ as a function.

$f (a) = e^{a x}$

Step 2

The function $F (a)$ can be found by finding the indefinite integral of the derivative $f (a)$ .

$F (a) = \int f (a) d a$

Step 3

Set up the integral to solve.

$F (a) = \int e^{a x} d a$

Step 4

Let $u = a x$ . Then $d u = x d a$ , so $\frac{1}{x} d u = d a$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 4.1

Let $u = a x$ . Find $\frac{d u}{d a}$ .

Tap for more steps...

Step 4.1.1

Differentiate $a x$ .

$\frac{d}{d a} [a x]$

Step 4.1.2

Since $x$ is constant with respect to $a$ , the derivative of $a x$ with respect to $a$ is $x \frac{d}{d a} [a]$ .

$x \frac{d}{d a} [a]$

Step 4.1.3

Differentiate using the Power Rule which states that $\frac{d}{d a} [a^{n}]$ is $n a^{n - 1}$ where $n = 1$ .

$x \cdot 1$

Step 4.1.4

Multiply $x$ by $1$ .

$x$

Step 4.2

Rewrite the problem using $u$ and $d u$ .

$\int e^{u} \frac{1}{x} d u$

Step 5

Combine $e^{u}$ and $\frac{1}{x}$ .

$\int \frac{e^{u}}{x} d u$

Step 6

Since $\frac{1}{x}$ is constant with respect to $u$ , move $\frac{1}{x}$ out of the integral.

$\frac{1}{x} \int e^{u} d u$

Step 7

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$\frac{1}{x} (e^{u} + C)$

Step 8

Simplify.

Tap for more steps...

Step 8.1

Simplify.

$\frac{1}{x} e^{u} + C$

Step 8.2

Combine $\frac{1}{x}$ and $e^{u}$ .

$\frac{e^{u}}{x} + C$

Step 9

Replace all occurrences of $u$ with $a x$ .

$\frac{e^{a x}}{x} + C$

Step 10

The answer is the antiderivative of the function $f (a) = e^{a x}$ .

$F (a) =$ $\frac{e^{a x}}{x} + C$