Calculus Examples

$\int 4 e^{2 x} d x$

Step 1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 \int e^{2 x} d x$

Step 2

Let $u = 2 x$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 2 x$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $2 x$ .

$\frac{d}{d x} [2 x]$

Step 2.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x]$

Step 2.1.3

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

$2 \cdot 1$

Step 2.1.4

Multiply $2$ by $1$ .

$2$

Step 2.2

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

$4 \int e^{u} \frac{1}{2} d u$

Step 3

Combine $e^{u}$ and $\frac{1}{2}$ .

$4 \int \frac{e^{u}}{2} d u$

Step 4

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$4 (\frac{1}{2} \int e^{u} d u)$

Step 5

Simplify.

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Step 5.1

Combine $\frac{1}{2}$ and $4$ .

$\frac{4}{2} \int e^{u} d u$

Step 5.2

Cancel the common factor of $4$ and $2$ .

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Step 5.2.1

Factor $2$ out of $4$ .

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

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $2$ out of $2$ .

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

Step 5.2.2.2

Cancel the common factor.

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

Step 5.2.2.3

Rewrite the expression.

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

Step 5.2.2.4

Divide $2$ by $1$ .

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

Step 6

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

$2 (e^{u} + C)$

Step 7

Simplify.

$2 e^{u} + C$

Step 8

Replace all occurrences of $u$ with $2 x$ .

$2 e^{2 x} + C$