Calculus Examples

$\int e^{5 x} d x$

Step 1

Let

$u = 5 x$ . Then

$d u = 5 d x$ , so

$\frac{1}{5} d u = d x$ . Rewrite using

$u$ and

$d$

$u$ .

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Step 1.1

Let

$u = 5 x$ . Find

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

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Step 1.1.1

Differentiate

$5 x$ .

$\frac{d}{d x} [5 x]$

Step 1.1.2

Since

$5$ is constant with respect to

$x$ , the derivative of

$5 x$ with respect to

$x$ is

$5 \frac{d}{d x} [x]$ .

$5 \frac{d}{d x} [x]$

Step 1.1.3

Differentiate using the Power Rule which states that

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

$n x^{n - 1}$ where

$n = 1$ .

$5 \cdot 1$

Step 1.1.4

Multiply

$5$ by

$1$ .

$5$

$5$

Step 1.2

Rewrite the problem using

$u$ and

$d u$ .

$\int e^{u} \frac{1}{5} d u$

$\int e^{u} \frac{1}{5} d u$

Step 2

Combine

$e^{u}$ and

$\frac{1}{5}$ .

$\int \frac{e^{u}}{5} d u$

Step 3

Since

$\frac{1}{5}$ is constant with respect to

$u$ , move

$\frac{1}{5}$ out of the integral.

$\frac{1}{5} \int e^{u} d u$

Step 4

The integral of

$e^{u}$ with respect to

$u$ is

$e^{u}$ .

$\frac{1}{5} (e^{u} + C)$

Step 5

Simplify.

$\frac{1}{5} e^{u} + C$

Step 6

Replace all occurrences of

$u$ with

$5 x$ .

$\frac{1}{5} e^{5 x} + C$

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