Calculus Examples

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

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = 4 x$ and $d v = e^{2 x}$ .

$4 x (\frac{1}{2} e^{2 x}) - \int \frac{1}{2} e^{2 x} \cdot 4 d x$

Step 2

Simplify.

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Step 2.1

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

$4 x \frac{e^{2 x}}{2} - \int \frac{1}{2} e^{2 x} \cdot 4 d x$

Step 2.2

Combine $4$ and $\frac{e^{2 x}}{2}$ .

$x \frac{4 e^{2 x}}{2} - \int \frac{1}{2} e^{2 x} \cdot 4 d x$

Step 2.3

Combine $x$ and $\frac{4 e^{2 x}}{2}$ .

$\frac{x (4 e^{2 x})}{2} - \int \frac{1}{2} e^{2 x} \cdot 4 d x$

Step 2.4

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

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Step 2.4.1

Factor $2$ out of $x (4 e^{2 x})$ .

$\frac{2 (x (2 e^{2 x}))}{2} - \int \frac{1}{2} e^{2 x} \cdot 4 d x$

Step 2.4.2

Cancel the common factors.

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Step 2.4.2.1

Factor $2$ out of $2$ .

$\frac{2 (x (2 e^{2 x}))}{2 (1)} - \int \frac{1}{2} e^{2 x} \cdot 4 d x$

Step 2.4.2.2

Cancel the common factor.

$\frac{2 (x (2 e^{2 x}))}{2 \cdot 1} - \int \frac{1}{2} e^{2 x} \cdot 4 d x$

Step 2.4.2.3

Rewrite the expression.

$\frac{x (2 e^{2 x})}{1} - \int \frac{1}{2} e^{2 x} \cdot 4 d x$

Step 2.4.2.4

Divide $x (2 e^{2 x})$ by $1$ .

$x (2 e^{2 x}) - \int \frac{1}{2} e^{2 x} \cdot 4 d x$

Step 3

Since $\frac{1}{2} \cdot 4$ is constant with respect to $x$ , move $\frac{1}{2} \cdot 4$ out of the integral.

$x (2 e^{2 x}) - (\frac{1}{2} \cdot 4 \int e^{2 x} d x)$

Step 4

Simplify.

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Step 4.1

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

$x (2 e^{2 x}) - (\frac{4}{2} \int e^{2 x} d x)$

Step 4.2

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

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Step 4.2.1

Factor $2$ out of $4$ .

$x (2 e^{2 x}) - (\frac{2 \cdot 2}{2} \int e^{2 x} d x)$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $2$ out of $2$ .

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

Step 4.2.2.2

Cancel the common factor.

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

Step 4.2.2.3

Rewrite the expression.

$x (2 e^{2 x}) - (\frac{2}{1} \int e^{2 x} d x)$

Step 4.2.2.4

Divide $2$ by $1$ .

$x (2 e^{2 x}) - (2 \int e^{2 x} d x)$

Step 4.3

Multiply $2$ by $- 1$ .

$x (2 e^{2 x}) - 2 \int e^{2 x} d x$

Step 5

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 5.1

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

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Step 5.1.1

Differentiate $2 x$ .

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

Step 5.1.2

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 5.1.3

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 5.1.4

Multiply $2$ by $1$ .

$2$

Step 5.2

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

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

Step 6

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

$x (2 e^{2 x}) - 2 \int \frac{e^{u}}{2} d u$

Step 7

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

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

Step 8

Simplify.

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Step 8.1

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

$x (2 e^{2 x}) + \frac{- 2}{2} \int e^{u} d u$

Step 8.2

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

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Step 8.2.1

Factor $2$ out of $- 2$ .

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

Step 8.2.2

Cancel the common factors.

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Step 8.2.2.1

Factor $2$ out of $2$ .

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

Step 8.2.2.2

Cancel the common factor.

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

Step 8.2.2.3

Rewrite the expression.

$x (2 e^{2 x}) + \frac{- 1}{1} \int e^{u} d u$

Step 8.2.2.4

Divide $- 1$ by $1$ .

$x (2 e^{2 x}) - \int e^{u} d u$

Step 9

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

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

Step 10

Rewrite $x (2 e^{2 x}) - (e^{u} + C)$ as $2 x e^{2 x} - e^{u} + C$ .

$2 x e^{2 x} - e^{u} + C$

Step 11

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

$2 x e^{2 x} - e^{2 x} + C$