Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Simplify.

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Step 3.1

Combine $\frac{1}{4}$ and $e^{4 x}$ .

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

Step 3.2

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

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

Step 3.3

Combine $\frac{1}{4}$ and $e^{4 x}$ .

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

Step 4

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

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

Step 5

Let $u = 4 x$ . Then $d u = 4 d x$ , so $\frac{1}{4} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = 4 x$ . Find $\frac{d u}{d x}$ .

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Step 5.1.1

Differentiate $4 x$ .

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

Step 5.1.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

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

Step 5.1.3

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

$4 \cdot 1$

Step 5.1.4

Multiply $4$ by $1$ .

$4$

Step 5.2

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

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

Step 6

Combine $e^{u}$ and $\frac{1}{4}$ .

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Multiply $\frac{1}{4}$ by $\frac{1}{4}$ .

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

Step 8.2

Multiply $4$ by $4$ .

$2 (\frac{x e^{4 x}}{4} - \frac{1}{16} \int e^{u} d u)$

Step 9

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

$2 (\frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u} + C))$

Step 10

Rewrite $2 (\frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u} + C))$ as $2 (\frac{1}{4} x e^{4 x} - \frac{1}{16} e^{u}) + C$ .

$2 (\frac{1}{4} x e^{4 x} - \frac{1}{16} e^{u}) + C$

Step 11

Replace all occurrences of $u$ with $4 x$ .

$2 (\frac{1}{4} x e^{4 x} - \frac{1}{16} e^{4 x}) + C$

Step 12

Simplify.

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Step 12.1

Simplify each term.

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Step 12.1.1

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

$2 (\frac{x}{4} e^{4 x} - \frac{1}{16} e^{4 x}) + C$

Step 12.1.2

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

$2 (\frac{x e^{4 x}}{4} - \frac{1}{16} e^{4 x}) + C$

Step 12.1.3

Combine $e^{4 x}$ and $\frac{1}{16}$ .

$2 (\frac{x e^{4 x}}{4} - \frac{e^{4 x}}{16}) + C$

Step 12.2

Apply the distributive property.

$2 \frac{x e^{4 x}}{4} + 2 (- \frac{e^{4 x}}{16}) + C$

Step 12.3

Cancel the common factor of $2$ .

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Step 12.3.1

Factor $2$ out of $4$ .

$2 \frac{x e^{4 x}}{2 (2)} + 2 (- \frac{e^{4 x}}{16}) + C$

Step 12.3.2

Cancel the common factor.

$2 \frac{x e^{4 x}}{2 \cdot 2} + 2 (- \frac{e^{4 x}}{16}) + C$

Step 12.3.3

Rewrite the expression.

$\frac{x e^{4 x}}{2} + 2 (- \frac{e^{4 x}}{16}) + C$

Step 12.4

Cancel the common factor of $2$ .

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Step 12.4.1

Move the leading negative in $- \frac{e^{4 x}}{16}$ into the numerator.

$\frac{x e^{4 x}}{2} + 2 \frac{- e^{4 x}}{16} + C$

Step 12.4.2

Factor $2$ out of $16$ .

$\frac{x e^{4 x}}{2} + 2 \frac{- e^{4 x}}{2 (8)} + C$

Step 12.4.3

Cancel the common factor.

$\frac{x e^{4 x}}{2} + 2 \frac{- e^{4 x}}{2 \cdot 8} + C$

Step 12.4.4

Rewrite the expression.

$\frac{x e^{4 x}}{2} + \frac{- e^{4 x}}{8} + C$

Step 12.5

Move the negative in front of the fraction.

$\frac{x e^{4 x}}{2} - \frac{e^{4 x}}{8} + C$

Step 13

Reorder terms.

$\frac{1}{2} x e^{4 x} - \frac{1}{8} e^{4 x} + C$