Calculus Examples

$\frac{d y}{d x} + 4 y = x - 2 x^{2}$

Step 1

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = 4$ .

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Step 1.1

Set up the integration.

$e^{\int 4 d x}$

Step 1.2

Apply the constant rule.

$e^{4 x + C}$

Step 1.3

Remove the constant of integration.

$e^{4 x}$

Step 2

Multiply each term by the integrating factor $e^{4 x}$ .

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Step 2.1

Multiply each term by $e^{4 x}$ .

$e^{4 x} \frac{d y}{d x} + e^{4 x} (4 y) = e^{4 x} x + e^{4 x} (- 2 x^{2})$

Step 2.2

Rewrite using the commutative property of multiplication.

$e^{4 x} \frac{d y}{d x} + 4 e^{4 x} y = e^{4 x} x + e^{4 x} (- 2 x^{2})$

Step 2.3

Rewrite using the commutative property of multiplication.

$e^{4 x} \frac{d y}{d x} + 4 e^{4 x} y = e^{4 x} x - 2 e^{4 x} x^{2}$

Step 2.4

Reorder factors in $e^{4 x} \frac{d y}{d x} + 4 e^{4 x} y = e^{4 x} x - 2 e^{4 x} x^{2}$ .

$e^{4 x} \frac{d y}{d x} + 4 y e^{4 x} = x e^{4 x} - 2 x^{2} e^{4 x}$

Step 3

Rewrite the left side as a result of differentiating a product.

$\frac{d}{d x} [e^{4 x} y] = x e^{4 x} - 2 x^{2} e^{4 x}$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [e^{4 x} y] d x = \int x e^{4 x} - 2 x^{2} e^{4 x} d x$

Step 5

Integrate the left side.

$e^{4 x} y = \int x e^{4 x} - 2 x^{2} e^{4 x} d x$

Step 6

Integrate the right side.

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Step 6.1

Split the single integral into multiple integrals.

$e^{4 x} y = \int x e^{4 x} d x + \int - 2 x^{2} e^{4 x} d x$

Step 6.2

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

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

Step 6.3

Simplify.

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Step 6.3.1

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

$e^{4 x} y = x \frac{e^{4 x}}{4} - \int \frac{1}{4} e^{4 x} d x + \int - 2 x^{2} e^{4 x} d x$

Step 6.3.2

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

$e^{4 x} y = \frac{x e^{4 x}}{4} - \int \frac{1}{4} e^{4 x} d x + \int - 2 x^{2} e^{4 x} d x$

Step 6.4

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

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

Step 6.5

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

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Step 6.5.1

Let $u_{1} = 4 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 6.5.1.1

Differentiate $4 x$ .

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

Step 6.5.1.2

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 6.5.1.3

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 6.5.1.4

Multiply $4$ by $1$ .

$4$

Step 6.5.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{4} \int e^{u_{1}} \frac{1}{4} d u_{1} + \int - 2 x^{2} e^{4 x} d x$

Step 6.6

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

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{4} \int \frac{e^{u_{1}}}{4} d u_{1} + \int - 2 x^{2} e^{4 x} d x$

Step 6.7

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

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

Step 6.8

Simplify.

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Step 6.8.1

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

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{4 \cdot 4} \int e^{u_{1}} d u_{1} + \int - 2 x^{2} e^{4 x} d x$

Step 6.8.2

Multiply $4$ by $4$ .

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{16} \int e^{u_{1}} d u_{1} + \int - 2 x^{2} e^{4 x} d x$

Step 6.9

The integral of $e^{u_{1}}$ with respect to $u_{1}$ is $e^{u_{1}}$ .

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) + \int - 2 x^{2} e^{4 x} d x$

Step 6.10

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

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) - 2 \int x^{2} e^{4 x} d x$

Step 6.11

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

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) - 2 (x^{2} (\frac{1}{4} e^{4 x}) - \int \frac{1}{4} e^{4 x} (2 x) d x)$

Step 6.12

Simplify.

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Step 6.12.1

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

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) - 2 (x^{2} \frac{e^{4 x}}{4} - \int \frac{1}{4} e^{4 x} (2 x) d x)$

Step 6.12.2

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

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) - 2 (\frac{x^{2} e^{4 x}}{4} - \int \frac{1}{4} e^{4 x} (2 x) d x)$

Step 6.12.3

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

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) - 2 (\frac{x^{2} e^{4 x}}{4} - \int \frac{e^{4 x}}{4} (2 x) d x)$

Step 6.12.4

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

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) - 2 (\frac{x^{2} e^{4 x}}{4} - \int \frac{2 e^{4 x}}{4} x d x)$

Step 6.12.5

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

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) - 2 (\frac{x^{2} e^{4 x}}{4} - \int \frac{2 e^{4 x} x}{4} d x)$

Step 6.12.6

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

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Step 6.12.6.1

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

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) - 2 (\frac{x^{2} e^{4 x}}{4} - \int \frac{2 (e^{4 x} x)}{4} d x)$

Step 6.12.6.2

Cancel the common factors.

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Step 6.12.6.2.1

Factor $2$ out of $4$ .

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) - 2 (\frac{x^{2} e^{4 x}}{4} - \int \frac{2 (e^{4 x} x)}{2 \cdot 2} d x)$

Step 6.12.6.2.2

Cancel the common factor.

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) - 2 (\frac{x^{2} e^{4 x}}{4} - \int \frac{2 (e^{4 x} x)}{2 \cdot 2} d x)$

Step 6.12.6.2.3

Rewrite the expression.

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) - 2 (\frac{x^{2} e^{4 x}}{4} - \int \frac{e^{4 x} x}{2} d x)$

Step 6.13

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

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) - 2 (\frac{x^{2} e^{4 x}}{4} - (\frac{1}{2} \int e^{4 x} x d x))$

Step 6.14

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

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) - 2 (\frac{x^{2} e^{4 x}}{4} - \frac{1}{2} (x (\frac{1}{4} e^{4 x}) - \int \frac{1}{4} e^{4 x} d x))$

Step 6.15

Simplify.

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Step 6.15.1

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

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) - 2 (\frac{x^{2} e^{4 x}}{4} - \frac{1}{2} (x \frac{e^{4 x}}{4} - \int \frac{1}{4} e^{4 x} d x))$

Step 6.15.2

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

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) - 2 (\frac{x^{2} e^{4 x}}{4} - \frac{1}{2} (\frac{x e^{4 x}}{4} - \int \frac{1}{4} e^{4 x} d x))$

Step 6.15.3

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

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) - 2 (\frac{x^{2} e^{4 x}}{4} - \frac{1}{2} (\frac{x e^{4 x}}{4} - \int \frac{e^{4 x}}{4} d x))$

Step 6.16

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

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) - 2 (\frac{x^{2} e^{4 x}}{4} - \frac{1}{2} (\frac{x e^{4 x}}{4} - (\frac{1}{4} \int e^{4 x} d x)))$

Step 6.17

Let $u_{2} = 4 x$ . Then $d u_{2} = 4 d x$ , so $\frac{1}{4} d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 6.17.1

Let $u_{2} = 4 x$ . Find $\frac{d u_{2}}{d x}$ .

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Step 6.17.1.1

Differentiate $4 x$ .

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

Step 6.17.1.2

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 6.17.1.3

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 6.17.1.4

Multiply $4$ by $1$ .

$4$

Step 6.17.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) - 2 (\frac{x^{2} e^{4 x}}{4} - \frac{1}{2} (\frac{x e^{4 x}}{4} - \frac{1}{4} \int e^{u_{2}} \frac{1}{4} d u_{2}))$

Step 6.18

Combine $e^{u_{2}}$ and $\frac{1}{4}$ .

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) - 2 (\frac{x^{2} e^{4 x}}{4} - \frac{1}{2} (\frac{x e^{4 x}}{4} - \frac{1}{4} \int \frac{e^{u_{2}}}{4} d u_{2}))$

Step 6.19

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

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) - 2 (\frac{x^{2} e^{4 x}}{4} - \frac{1}{2} (\frac{x e^{4 x}}{4} - \frac{1}{4} (\frac{1}{4} \int e^{u_{2}} d u_{2})))$

Step 6.20

Simplify.

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Step 6.20.1

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

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) - 2 (\frac{x^{2} e^{4 x}}{4} - \frac{1}{2} (\frac{x e^{4 x}}{4} - \frac{1}{4 \cdot 4} \int e^{u_{2}} d u_{2}))$

Step 6.20.2

Multiply $4$ by $4$ .

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C) - 2 (\frac{x^{2} e^{4 x}}{4} - \frac{1}{2} (\frac{x e^{4 x}}{4} - \frac{1}{16} \int e^{u_{2}} d u_{2}))$

Step 6.21

The integral of $e^{u_{2}}$ with respect to $u_{2}$ is $e^{u_{2}}$ .

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

Step 6.22

Simplify.

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Step 6.22.1

Simplify.

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

Step 6.22.2

Simplify.

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Step 6.22.2.1

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

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

Step 6.22.2.2

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

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

Step 6.22.2.3

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

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

Step 6.22.2.4

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

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

Step 6.22.2.5

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

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

Step 6.22.2.6

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

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

Step 6.22.2.7

Combine $e^{u_{2}}$ and $\frac{1}{16}$ .

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

Step 6.22.2.8

To write $- \frac{1}{2} (\frac{x e^{4 x}}{4} - \frac{e^{u_{2}}}{16})$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 6.22.2.9

Combine $- \frac{1}{2} (\frac{x e^{4 x}}{4} - \frac{e^{u_{2}}}{16})$ and $\frac{4}{4}$ .

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

Step 6.22.2.10

Combine the numerators over the common denominator.

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

Step 6.22.2.11

Multiply $4$ by $- 1$ .

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

Step 6.22.2.12

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

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

Step 6.22.2.13

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

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Step 6.22.2.13.1

Factor $2$ out of $- 4$ .

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

Step 6.22.2.13.2

Cancel the common factors.

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Step 6.22.2.13.2.1

Factor $2$ out of $2$ .

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

Step 6.22.2.13.2.2

Cancel the common factor.

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

Step 6.22.2.13.2.3

Rewrite the expression.

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

Step 6.22.2.13.2.4

Divide $- 2$ by $1$ .

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

Step 6.22.2.14

Combine $- 2$ and $\frac{x^{2} e^{4 x} - 2 (\frac{x e^{4 x}}{4} - \frac{e^{u_{2}}}{16})}{4}$ .

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

Step 6.22.2.15

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

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Step 6.22.2.15.1

Factor $2$ out of $- 2 (x^{2} e^{4 x} - 2 (\frac{x e^{4 x}}{4} - \frac{e^{u_{2}}}{16}))$ .

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

Step 6.22.2.15.2

Cancel the common factors.

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Step 6.22.2.15.2.1

Factor $2$ out of $4$ .

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

Step 6.22.2.15.2.2

Cancel the common factor.

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

Step 6.22.2.15.2.3

Rewrite the expression.

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

Step 6.22.2.16

Move the negative in front of the fraction.

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

Step 6.23

Substitute back in for each integration substitution variable.

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Step 6.23.1

Replace all occurrences of $u_{1}$ with $4 x$ .

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

Step 6.23.2

Replace all occurrences of $u_{2}$ with $4 x$ .

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

Step 6.24

Simplify.

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Step 6.24.1

Apply the distributive property.

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

Step 6.24.2

Cancel the common factor of $2$ .

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Step 6.24.2.1

Factor $2$ out of $- 2$ .

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

Step 6.24.2.2

Factor $2$ out of $4$ .

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

Step 6.24.2.3

Cancel the common factor.

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

Step 6.24.2.4

Rewrite the expression.

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

Step 6.24.3

Cancel the common factor of $2$ .

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Step 6.24.3.1

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

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{16} e^{4 x} - \frac{x^{2} e^{4 x} - \frac{x e^{4 x}}{2} - 2 \frac{- e^{4 x}}{16}}{2} + C$

Step 6.24.3.2

Factor $2$ out of $- 2$ .

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

Step 6.24.3.3

Factor $2$ out of $16$ .

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{16} e^{4 x} - \frac{x^{2} e^{4 x} - \frac{x e^{4 x}}{2} + 2 \cdot - 1 \frac{- e^{4 x}}{2 \cdot 8}}{2} + C$

Step 6.24.3.4

Cancel the common factor.

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{16} e^{4 x} - \frac{x^{2} e^{4 x} - \frac{x e^{4 x}}{2} + 2 \cdot - 1 \frac{- e^{4 x}}{2 \cdot 8}}{2} + C$

Step 6.24.3.5

Rewrite the expression.

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{16} e^{4 x} - \frac{x^{2} e^{4 x} - \frac{x e^{4 x}}{2} - \frac{- e^{4 x}}{8}}{2} + C$

Step 6.24.4

Simplify each term.

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Step 6.24.4.1

Move the negative in front of the fraction.

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{16} e^{4 x} - \frac{x^{2} e^{4 x} - \frac{x e^{4 x}}{2} - - \frac{e^{4 x}}{8}}{2} + C$

Step 6.24.4.2

Multiply $- - \frac{e^{4 x}}{8}$ .

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Step 6.24.4.2.1

Multiply $- 1$ by $- 1$ .

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{16} e^{4 x} - \frac{x^{2} e^{4 x} - \frac{x e^{4 x}}{2} + 1 \frac{e^{4 x}}{8}}{2} + C$

Step 6.24.4.2.2

Multiply $\frac{e^{4 x}}{8}$ by $1$ .

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{1}{16} e^{4 x} - \frac{x^{2} e^{4 x} - \frac{x e^{4 x}}{2} + \frac{e^{4 x}}{8}}{2} + C$

Step 6.24.5

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

$e^{4 x} y = \frac{x e^{4 x}}{4} - \frac{e^{4 x}}{16} - \frac{x^{2} e^{4 x} - \frac{x e^{4 x}}{2} + \frac{e^{4 x}}{8}}{2} + C$

Step 6.25

Reorder terms.

$e^{4 x} y = \frac{1}{4} x e^{4 x} - \frac{1}{16} e^{4 x} - \frac{1}{2} (x^{2} e^{4 x} - \frac{1}{2} x e^{4 x} + \frac{1}{8} e^{4 x}) + C$

Step 7

Solve for $y$ .

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Step 7.1

Simplify.

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Step 7.1.1

Simplify each term.

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Step 7.1.1.1

Multiply $- \frac{1}{2} (x e^{4 x})$ .

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Step 7.1.1.1.1

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

$e^{4 x} y = \frac{1}{4} \cdot (x e^{4 x}) - \frac{1}{16} \cdot e^{4 x} - \frac{1}{2} \cdot (x^{2} e^{4 x} - \frac{x}{2} e^{4 x} + \frac{1}{8} \cdot e^{4 x}) + C$

Step 7.1.1.1.2

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

$e^{4 x} y = \frac{1}{4} \cdot (x e^{4 x}) - \frac{1}{16} \cdot e^{4 x} - \frac{1}{2} \cdot (x^{2} e^{4 x} - \frac{e^{4 x} x}{2} + \frac{1}{8} \cdot e^{4 x}) + C$

Step 7.1.1.2

Combine $\frac{1}{8}$ and $e^{4 x}$ .

$e^{4 x} y = \frac{1}{4} \cdot (x e^{4 x}) - \frac{1}{16} \cdot e^{4 x} - \frac{1}{2} \cdot (x^{2} e^{4 x} - \frac{e^{4 x} x}{2} + \frac{e^{4 x}}{8}) + C$

Step 7.1.2

Reorder the factors of $x^{2} e^{4 x}$ .

$e^{4 x} y = \frac{1}{4} \cdot (x e^{4 x}) - \frac{1}{16} \cdot e^{4 x} - \frac{1}{2} \cdot (e^{4 x} x^{2} - \frac{e^{4 x} x}{2} + \frac{e^{4 x}}{8}) + C$

Step 7.1.3

Add $e^{4 x} x^{2}$ and $\frac{e^{4 x}}{8}$ .

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Step 7.1.3.1

Reorder $e^{4 x}$ and $x^{2}$ .

$e^{4 x} y = \frac{1}{4} \cdot (x e^{4 x}) - \frac{1}{16} \cdot e^{4 x} - \frac{1}{2} \cdot (- \frac{e^{4 x} x}{2} + x^{2} e^{4 x} + \frac{e^{4 x}}{8}) + C$

Step 7.1.3.2

To write $x^{2} e^{4 x}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$e^{4 x} y = \frac{1}{4} \cdot (x e^{4 x}) - \frac{1}{16} \cdot e^{4 x} - \frac{1}{2} \cdot (- \frac{e^{4 x} x}{2} + x^{2} e^{4 x} \cdot \frac{8}{8} + \frac{e^{4 x}}{8}) + C$

Step 7.1.3.3

Combine $x^{2} e^{4 x}$ and $\frac{8}{8}$ .

$e^{4 x} y = \frac{1}{4} \cdot (x e^{4 x}) - \frac{1}{16} \cdot e^{4 x} - \frac{1}{2} \cdot (- \frac{e^{4 x} x}{2} + \frac{x^{2} e^{4 x} \cdot 8}{8} + \frac{e^{4 x}}{8}) + C$

Step 7.1.3.4

Combine the numerators over the common denominator.

$e^{4 x} y = \frac{1}{4} \cdot (x e^{4 x}) - \frac{1}{16} \cdot e^{4 x} - \frac{1}{2} \cdot (- \frac{e^{4 x} x}{2} + \frac{x^{2} e^{4 x} \cdot 8 + e^{4 x}}{8}) + C$

Step 7.1.4

Simplify the numerator.

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Step 7.1.4.1

Factor $e^{4 x}$ out of $x^{2} e^{4 x} \cdot 8 + e^{4 x}$ .

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Step 7.1.4.1.1

Factor $e^{4 x}$ out of $x^{2} e^{4 x} \cdot 8$ .

$e^{4 x} y = \frac{1}{4} \cdot (x e^{4 x}) - \frac{1}{16} \cdot e^{4 x} - \frac{1}{2} \cdot (- \frac{e^{4 x} x}{2} + \frac{e^{4 x} (x^{2} \cdot 8) + e^{4 x}}{8}) + C$

Step 7.1.4.1.2

Multiply by $1$ .

$e^{4 x} y = \frac{1}{4} \cdot (x e^{4 x}) - \frac{1}{16} \cdot e^{4 x} - \frac{1}{2} \cdot (- \frac{e^{4 x} x}{2} + \frac{e^{4 x} (x^{2} \cdot 8) + e^{4 x} \cdot 1}{8}) + C$

Step 7.1.4.1.3

Factor $e^{4 x}$ out of $e^{4 x} (x^{2} \cdot 8) + e^{4 x} \cdot 1$ .

$e^{4 x} y = \frac{1}{4} \cdot (x e^{4 x}) - \frac{1}{16} \cdot e^{4 x} - \frac{1}{2} \cdot (- \frac{e^{4 x} x}{2} + \frac{e^{4 x} (x^{2} \cdot 8 + 1)}{8}) + C$

Step 7.1.4.2

Move $8$ to the left of $x^{2}$ .

$e^{4 x} y = \frac{1}{4} \cdot (x e^{4 x}) - \frac{1}{16} \cdot e^{4 x} - \frac{1}{2} \cdot (- \frac{e^{4 x} x}{2} + \frac{e^{4 x} (8 x^{2} + 1)}{8}) + C$

Step 7.1.5

To write $- \frac{e^{4 x} x}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e^{4 x} y = \frac{1}{4} \cdot (x e^{4 x}) - \frac{1}{16} \cdot e^{4 x} - \frac{1}{2} \cdot (- \frac{e^{4 x} x}{2} \cdot \frac{4}{4} + \frac{e^{4 x} (8 x^{2} + 1)}{8}) + C$

Step 7.1.6

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.1.6.1

Multiply $\frac{e^{4 x} x}{2}$ by $\frac{4}{4}$ .

$e^{4 x} y = \frac{1}{4} \cdot (x e^{4 x}) - \frac{1}{16} \cdot e^{4 x} - \frac{1}{2} \cdot (- \frac{e^{4 x} x \cdot 4}{2 \cdot 4} + \frac{e^{4 x} (8 x^{2} + 1)}{8}) + C$

Step 7.1.6.2

Multiply $2$ by $4$ .

$e^{4 x} y = \frac{1}{4} \cdot (x e^{4 x}) - \frac{1}{16} \cdot e^{4 x} - \frac{1}{2} \cdot (- \frac{e^{4 x} x \cdot 4}{8} + \frac{e^{4 x} (8 x^{2} + 1)}{8}) + C$

Step 7.1.7

Combine the numerators over the common denominator.

$e^{4 x} y = \frac{1}{4} \cdot (x e^{4 x}) - \frac{1}{16} \cdot e^{4 x} - \frac{1}{2} \cdot \frac{- e^{4 x} x \cdot 4 + e^{4 x} (8 x^{2} + 1)}{8} + C$

Step 7.1.8

Simplify the numerator.

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Step 7.1.8.1

Factor $e^{4 x}$ out of $- e^{4 x} x \cdot 4 + e^{4 x} (8 x^{2} + 1)$ .

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Step 7.1.8.1.1

Factor $e^{4 x}$ out of $- e^{4 x} x \cdot 4$ .

$e^{4 x} y = \frac{1}{4} \cdot (x e^{4 x}) - \frac{1}{16} \cdot e^{4 x} - \frac{1}{2} \cdot \frac{e^{4 x} (- x \cdot 4) + e^{4 x} (8 x^{2} + 1)}{8} + C$

Step 7.1.8.1.2

Factor $e^{4 x}$ out of $e^{4 x} (- x \cdot 4) + e^{4 x} (8 x^{2} + 1)$ .

$e^{4 x} y = \frac{1}{4} \cdot (x e^{4 x}) - \frac{1}{16} \cdot e^{4 x} - \frac{1}{2} \cdot \frac{e^{4 x} (- x \cdot 4 + 8 x^{2} + 1)}{8} + C$

Step 7.1.8.2

Multiply $4$ by $- 1$ .

$e^{4 x} y = \frac{1}{4} \cdot (x e^{4 x}) - \frac{1}{16} \cdot e^{4 x} - \frac{1}{2} \cdot \frac{e^{4 x} (- 4 x + 8 x^{2} + 1)}{8} + C$

Step 7.1.8.3

Reorder terms.

$e^{4 x} y = \frac{1}{4} \cdot (x e^{4 x}) - \frac{1}{16} \cdot e^{4 x} - \frac{1}{2} \cdot \frac{e^{4 x} (8 x^{2} - 4 x + 1)}{8} + C$

Step 7.1.9

Multiply $- \frac{1}{2} \cdot \frac{e^{4 x} (8 x^{2} - 4 x + 1)}{8}$ .

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Step 7.1.9.1

Multiply $\frac{e^{4 x} (8 x^{2} - 4 x + 1)}{8}$ by $\frac{1}{2}$ .

$e^{4 x} y = \frac{1}{4} \cdot (x e^{4 x}) - \frac{1}{16} \cdot e^{4 x} - \frac{e^{4 x} (8 x^{2} - 4 x + 1)}{8 \cdot 2} + C$

Step 7.1.9.2

Multiply $8$ by $2$ .

$e^{4 x} y = \frac{1}{4} \cdot (x e^{4 x}) - \frac{1}{16} \cdot e^{4 x} - \frac{e^{4 x} (8 x^{2} - 4 x + 1)}{16} + C$

Step 7.1.10

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

$e^{4 x} y = \frac{1}{4} \cdot (x e^{4 x}) - \frac{e^{4 x}}{16} - \frac{e^{4 x} (8 x^{2} - 4 x + 1)}{16} + C$

Step 7.1.11

Remove parentheses.

$e^{4 x} y = \frac{1}{4} \cdot x e^{4 x} - \frac{e^{4 x}}{16} - \frac{e^{4 x} (8 x^{2} - 4 x + 1)}{16} + C$

Step 7.2

Divide each term in $e^{4 x} y = \frac{1}{4} \cdot (x e^{4 x}) - \frac{e^{4 x}}{16} - \frac{e^{4 x} (8 x^{2} - 4 x + 1)}{16} + C$ by $e^{4 x}$ and simplify.

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Step 7.2.1

Divide each term in $e^{4 x} y = \frac{1}{4} \cdot (x e^{4 x}) - \frac{e^{4 x}}{16} - \frac{e^{4 x} (8 x^{2} - 4 x + 1)}{16} + C$ by $e^{4 x}$ .

$\frac{e^{4 x} y}{e^{4 x}} = \frac{\frac{1}{4} \cdot (x e^{4 x})}{e^{4 x}} + \frac{- \frac{e^{4 x}}{16}}{e^{4 x}} + \frac{- \frac{e^{4 x} (8 x^{2} - 4 x + 1)}{16}}{e^{4 x}} + \frac{C}{e^{4 x}}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $e^{4 x}$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{e^{4 x} y}{e^{4 x}} = \frac{\frac{1}{4} \cdot (x e^{4 x})}{e^{4 x}} + \frac{- \frac{e^{4 x}}{16}}{e^{4 x}} + \frac{- \frac{e^{4 x} (8 x^{2} - 4 x + 1)}{16}}{e^{4 x}} + \frac{C}{e^{4 x}}$

Step 7.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{1}{4} \cdot (x e^{4 x})}{e^{4 x}} + \frac{- \frac{e^{4 x}}{16}}{e^{4 x}} + \frac{- \frac{e^{4 x} (8 x^{2} - 4 x + 1)}{16}}{e^{4 x}} + \frac{C}{e^{4 x}}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Combine fractions.

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Step 7.2.3.1.1

Combine the numerators over the common denominator.

$y = \frac{\frac{1}{4} \cdot (x e^{4 x}) - \frac{e^{4 x}}{16} - \frac{e^{4 x} (8 x^{2} - 4 x + 1)}{16} + C}{e^{4 x}}$

Step 7.2.3.1.2

Combine the numerators over the common denominator.

$y = \frac{\frac{1}{4} \cdot (x e^{4 x}) + C + \frac{- e^{4 x} - e^{4 x} (8 x^{2} - 4 x + 1)}{16}}{e^{4 x}}$

Step 7.2.3.2

Simplify each term.

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Step 7.2.3.2.1

Apply the distributive property.

$y = \frac{\frac{1}{4} \cdot (x e^{4 x}) + C + \frac{- e^{4 x} - e^{4 x} (8 x^{2}) - e^{4 x} (- 4 x) - e^{4 x} \cdot 1}{16}}{e^{4 x}}$

Step 7.2.3.2.2

Simplify.

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Step 7.2.3.2.2.1

Rewrite using the commutative property of multiplication.

$y = \frac{\frac{1}{4} \cdot (x e^{4 x}) + C + \frac{- e^{4 x} - 1 \cdot 8 e^{4 x} x^{2} - e^{4 x} (- 4 x) - e^{4 x} \cdot 1}{16}}{e^{4 x}}$

Step 7.2.3.2.2.2

Rewrite using the commutative property of multiplication.

$y = \frac{\frac{1}{4} \cdot (x e^{4 x}) + C + \frac{- e^{4 x} - 1 \cdot 8 e^{4 x} x^{2} - 1 \cdot - 4 e^{4 x} x - e^{4 x} \cdot 1}{16}}{e^{4 x}}$

Step 7.2.3.2.2.3

Multiply $- 1$ by $1$ .

$y = \frac{\frac{1}{4} \cdot (x e^{4 x}) + C + \frac{- e^{4 x} - 1 \cdot 8 e^{4 x} x^{2} - 1 \cdot - 4 e^{4 x} x - e^{4 x}}{16}}{e^{4 x}}$

Step 7.2.3.2.3

Simplify each term.

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Step 7.2.3.2.3.1

Multiply $- 1$ by $8$ .

$y = \frac{\frac{1}{4} \cdot (x e^{4 x}) + C + \frac{- e^{4 x} - 8 e^{4 x} x^{2} - 1 \cdot - 4 e^{4 x} x - e^{4 x}}{16}}{e^{4 x}}$

Step 7.2.3.2.3.2

Multiply $- 1$ by $- 4$ .

$y = \frac{\frac{1}{4} \cdot (x e^{4 x}) + C + \frac{- e^{4 x} - 8 e^{4 x} x^{2} + 4 e^{4 x} x - e^{4 x}}{16}}{e^{4 x}}$

Step 7.2.3.3

Simplify by adding terms.

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Step 7.2.3.3.1

Subtract $e^{4 x}$ from $- e^{4 x}$ .

$y = \frac{\frac{1}{4} \cdot (x e^{4 x}) + C + \frac{- 8 e^{4 x} x^{2} + 4 e^{4 x} x - 2 e^{4 x}}{16}}{e^{4 x}}$

Step 7.2.3.3.2

Reorder factors in $- 8 e^{4 x} x^{2} + 4 e^{4 x} x - 2 e^{4 x}$ .

$y = \frac{\frac{1}{4} \cdot (x e^{4 x}) + C + \frac{- 8 x^{2} e^{4 x} + 4 x e^{4 x} - 2 e^{4 x}}{16}}{e^{4 x}}$

Step 7.2.3.4

Simplify each term.

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Step 7.2.3.4.1

Multiply $\frac{1}{4} (x e^{4 x})$ .

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Step 7.2.3.4.1.1

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

$y = \frac{\frac{x}{4} e^{4 x} + C + \frac{- 8 x^{2} e^{4 x} + 4 x e^{4 x} - 2 e^{4 x}}{16}}{e^{4 x}}$

Step 7.2.3.4.1.2

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

$y = \frac{\frac{x e^{4 x}}{4} + C + \frac{- 8 x^{2} e^{4 x} + 4 x e^{4 x} - 2 e^{4 x}}{16}}{e^{4 x}}$

Step 7.2.3.4.2

Factor $2 e^{4 x}$ out of $- 8 x^{2} e^{4 x} + 4 x e^{4 x} - 2 e^{4 x}$ .

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Step 7.2.3.4.2.1

Factor $2 e^{4 x}$ out of $- 8 x^{2} e^{4 x}$ .

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

Step 7.2.3.4.2.2

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

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

Step 7.2.3.4.2.3

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

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

Step 7.2.3.4.2.4

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

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

Step 7.2.3.4.2.5

Factor $2 e^{4 x}$ out of $2 e^{4 x} (- 4 x^{2} + 2 x) + 2 e^{4 x} (- 1)$ .

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

Step 7.2.3.4.3

Cancel the common factor of $2$ and $16$ .

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Step 7.2.3.4.3.1

Factor $2$ out of $2 e^{4 x} (- 4 x^{2} + 2 x - 1)$ .

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

Step 7.2.3.4.3.2

Cancel the common factors.

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Step 7.2.3.4.3.2.1

Factor $2$ out of $16$ .

$y = \frac{\frac{x e^{4 x}}{4} + C + \frac{2 (e^{4 x} (- 4 x^{2} + 2 x - 1))}{2 \cdot 8}}{e^{4 x}}$

Step 7.2.3.4.3.2.2

Cancel the common factor.

$y = \frac{\frac{x e^{4 x}}{4} + C + \frac{2 (e^{4 x} (- 4 x^{2} + 2 x - 1))}{2 \cdot 8}}{e^{4 x}}$

Step 7.2.3.4.3.2.3

Rewrite the expression.

$y = \frac{\frac{x e^{4 x}}{4} + C + \frac{e^{4 x} (- 4 x^{2} + 2 x - 1)}{8}}{e^{4 x}}$

Step 7.2.3.5

To write $\frac{x e^{4 x}}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \frac{C + \frac{x e^{4 x}}{4} \cdot \frac{2}{2} + \frac{e^{4 x} (- 4 x^{2} + 2 x - 1)}{8}}{e^{4 x}}$

Step 7.2.3.6

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.2.3.6.1

Multiply $\frac{x e^{4 x}}{4}$ by $\frac{2}{2}$ .

$y = \frac{C + \frac{x e^{4 x} \cdot 2}{4 \cdot 2} + \frac{e^{4 x} (- 4 x^{2} + 2 x - 1)}{8}}{e^{4 x}}$

Step 7.2.3.6.2

Multiply $4$ by $2$ .

$y = \frac{C + \frac{x e^{4 x} \cdot 2}{8} + \frac{e^{4 x} (- 4 x^{2} + 2 x - 1)}{8}}{e^{4 x}}$

Step 7.2.3.7

Combine the numerators over the common denominator.

$y = \frac{C + \frac{x e^{4 x} \cdot 2 + e^{4 x} (- 4 x^{2} + 2 x - 1)}{8}}{e^{4 x}}$

Step 7.2.3.8

Simplify each term.

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Step 7.2.3.8.1

Simplify the numerator.

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Step 7.2.3.8.1.1

Factor $e^{4 x}$ out of $x e^{4 x} \cdot 2 + e^{4 x} (- 4 x^{2} + 2 x - 1)$ .

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Step 7.2.3.8.1.1.1

Factor $e^{4 x}$ out of $x e^{4 x} \cdot 2$ .

$y = \frac{C + \frac{e^{4 x} (x \cdot 2) + e^{4 x} (- 4 x^{2} + 2 x - 1)}{8}}{e^{4 x}}$

Step 7.2.3.8.1.1.2

Factor $e^{4 x}$ out of $e^{4 x} (x \cdot 2) + e^{4 x} (- 4 x^{2} + 2 x - 1)$ .

$y = \frac{C + \frac{e^{4 x} (x \cdot 2 - 4 x^{2} + 2 x - 1)}{8}}{e^{4 x}}$

Step 7.2.3.8.1.2

Reorder terms.

$y = \frac{C + \frac{e^{4 x} (- 4 x^{2} + 2 x + 2 x - 1)}{8}}{e^{4 x}}$

Step 7.2.3.8.1.3

Factor out the greatest common factor from each group.

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Step 7.2.3.8.1.3.1

Group the first two terms and the last two terms.

$y = \frac{C + \frac{e^{4 x} ((- 4 x^{2} + 2 x) + 2 x - 1)}{8}}{e^{4 x}}$

Step 7.2.3.8.1.3.2

Factor out the greatest common factor (GCF) from each group.

$y = \frac{C + \frac{e^{4 x} (2 x (- 2 x + 1) - (- 2 x + 1))}{8}}{e^{4 x}}$

Step 7.2.3.8.1.4

Factor the polynomial by factoring out the greatest common factor, $- 2 x + 1$ .

$y = \frac{C + \frac{e^{4 x} (- 2 x + 1) (2 x - 1)}{8}}{e^{4 x}}$

Step 7.2.3.8.1.5

Combine exponents.

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Step 7.2.3.8.1.5.1

Factor $- 1$ out of $- 2 x$ .

$y = \frac{C + \frac{e^{4 x} (- (2 x) + 1) (2 x - 1)}{8}}{e^{4 x}}$

Step 7.2.3.8.1.5.2

Rewrite $1$ as $- 1 (- 1)$ .

$y = \frac{C + \frac{e^{4 x} (- (2 x) - 1 (- 1)) (2 x - 1)}{8}}{e^{4 x}}$

Step 7.2.3.8.1.5.3

Factor $- 1$ out of $- (2 x) - 1 (- 1)$ .

$y = \frac{C + \frac{e^{4 x} (- (2 x - 1)) (2 x - 1)}{8}}{e^{4 x}}$

Step 7.2.3.8.1.5.4

Rewrite $- (2 x - 1)$ as $- 1 (2 x - 1)$ .

$y = \frac{C + \frac{e^{4 x} (- 1 (2 x - 1)) (2 x - 1)}{8}}{e^{4 x}}$

Step 7.2.3.8.1.5.5

Raise $2 x - 1$ to the power of $1$ .

$y = \frac{C + \frac{e^{4 x} \cdot - 1 ({(2 x - 1)}^{1} (2 x - 1))}{8}}{e^{4 x}}$

Step 7.2.3.8.1.5.6

Raise $2 x - 1$ to the power of $1$ .

$y = \frac{C + \frac{e^{4 x} \cdot - 1 ({(2 x - 1)}^{1} {(2 x - 1)}^{1})}{8}}{e^{4 x}}$

Step 7.2.3.8.1.5.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = \frac{C + \frac{e^{4 x} \cdot - 1 {(2 x - 1)}^{1 + 1}}{8}}{e^{4 x}}$

Step 7.2.3.8.1.5.8

Add $1$ and $1$ .

$y = \frac{C + \frac{e^{4 x} \cdot - 1 {(2 x - 1)}^{2}}{8}}{e^{4 x}}$

Step 7.2.3.8.1.6

Factor out negative.

$y = \frac{C + \frac{- e^{4 x} {(2 x - 1)}^{2}}{8}}{e^{4 x}}$

Step 7.2.3.8.2

Move the negative in front of the fraction.

$y = \frac{C - \frac{e^{4 x} {(2 x - 1)}^{2}}{8}}{e^{4 x}}$

Step 7.2.3.9

To write $C$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$y = \frac{C \cdot \frac{8}{8} - \frac{e^{4 x} {(2 x - 1)}^{2}}{8}}{e^{4 x}}$

Step 7.2.3.10

Simplify terms.

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Step 7.2.3.10.1

Combine $C$ and $\frac{8}{8}$ .

$y = \frac{\frac{C \cdot 8}{8} - \frac{e^{4 x} {(2 x - 1)}^{2}}{8}}{e^{4 x}}$

Step 7.2.3.10.2

Combine the numerators over the common denominator.

$y = \frac{\frac{C \cdot 8 - e^{4 x} {(2 x - 1)}^{2}}{8}}{e^{4 x}}$

Step 7.2.3.11

Move $8$ to the left of $C$ .

$y = \frac{\frac{8 C - e^{4 x} {(2 x - 1)}^{2}}{8}}{e^{4 x}}$

Step 7.2.3.12

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{8 C - e^{4 x} {(2 x - 1)}^{2}}{8} \cdot \frac{1}{e^{4 x}}$

Step 7.2.3.13

Multiply $\frac{8 C - e^{4 x} {(2 x - 1)}^{2}}{8}$ by $\frac{1}{e^{4 x}}$ .

$y = \frac{8 C - e^{4 x} {(2 x - 1)}^{2}}{8 e^{4 x}}$