Calculus Examples

$\int 2 e^{5 x} d x$

Step 1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int e^{5 x} d x$

Step 2

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 2.1

Let $u = 5 x$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $5 x$ .

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

Step 2.1.2

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 2.1.3

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 2.1.4

Multiply $5$ by $1$ .

$5$

Step 2.2

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

$2 \int e^{u} \frac{1}{5} d u$

Step 3

Combine $e^{u}$ and $\frac{1}{5}$ .

$2 \int \frac{e^{u}}{5} d u$

Step 4

Since $\frac{1}{5}$ is constant with respect to $u$ , move $\frac{1}{5}$ out of the integral.

$2 (\frac{1}{5} \int e^{u} d u)$

Step 5

Combine $\frac{1}{5}$ and $2$ .

$\frac{2}{5} \int e^{u} d u$

Step 6

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

$\frac{2}{5} (e^{u} + C)$

Step 7

Simplify.

$\frac{2}{5} e^{u} + C$

Step 8

Replace all occurrences of $u$ with $5 x$ .

$\frac{2}{5} e^{5 x} + C$