Calculus Examples

$e^{4 x}$

Step 1

Write $e^{4 x}$ as a function.

$f (x) = e^{4 x}$

Step 2

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

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

Step 3

Set up the integral to solve.

$F (x) = \int e^{4 x} d x$

Step 4

Let $u = 4 x$ . Then $d u = 4 d x$ , so $\frac{1}{4} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 4.1

Let $u = 4 x$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 4.1.1

Differentiate $4 x$ .

$\frac{d}{d x} [4 x]$

Step 4.1.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$4 \frac{d}{d x} [x]$

Step 4.1.3

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

$4 \cdot 1$

Step 4.1.4

Multiply $4$ by $1$ .

$4$

Step 4.2

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

$\int e^{u} \frac{1}{4} d u$

Step 5

Combine $e^{u}$ and $\frac{1}{4}$ .

$\int \frac{e^{u}}{4} d u$

Step 6

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int e^{u} d u$

Step 7

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

$\frac{1}{4} (e^{u} + C)$

Step 8

Simplify.

$\frac{1}{4} e^{u} + C$

Step 9

Replace all occurrences of $u$ with $4 x$ .

$\frac{1}{4} e^{4 x} + C$

Step 10

The answer is the antiderivative of the function $f (x) = e^{4 x}$ .

$F (x) =$ $\frac{1}{4} e^{4 x} + C$