Calculus Examples

\int e^{4 x} d x

$\int e^{4 x} d x$

Step 1

Let

u = 4 x

$u = 4 x$ . Then

d u = 4 d x

$d u = 4 d x$ , so

\frac{1}{4} d u = d x

$\frac{1}{4} d u = d x$ . Rewrite using

u

$u$ and

d

$d$

u

$u$ .

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Step 1.1

Let

u = 4 x

$u = 4 x$ . Find

\frac{d u}{d x}

$\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate

4 x

$4 x$ .

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

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

Step 1.1.2

Since

4

$4$ is constant with respect to

x

$x$ , the derivative of

4 x

$4 x$ with respect to

x

$x$ is

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

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

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

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

Step 1.1.3

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

4 \cdot 1

$4 \cdot 1$

Step 1.1.4

Multiply

4

$4$ by

1

$1$ .

4

$4$

4

$4$

Step 1.2

Rewrite the problem using

u

$u$ and

d u

$d u$ .

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

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

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

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

Step 2

Combine

e^{u}

$e^{u}$ and

\frac{1}{4}

$\frac{1}{4}$ .

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

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

Step 3

Since

\frac{1}{4}

$\frac{1}{4}$ is constant with respect to

u

$u$ , move

\frac{1}{4}

$\frac{1}{4}$ out of the integral.

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

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

Step 4

The integral of

e^{u}

$e^{u}$ with respect to

u

$u$ is

e^{u}

$e^{u}$ .

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

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

Step 5

Simplify.

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

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

Step 6

Replace all occurrences of

u

$u$ with

4 x

$4 x$ .

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

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

[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$