Calculus Examples

$\frac{d y}{d x} + 4 y = 7 x + 9$

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 $e^{4 x}$ .

$e^{4 x} \frac{d y}{d x} + e^{4 x} (4 y) = e^{4 x} (7 x) + e^{4 x} \cdot 9$

Step 2.2

Rewrite using the commutative property of multiplication.

$e^{4 x} \frac{d y}{d x} + 4 e^{4 x} y = e^{4 x} (7 x) + e^{4 x} \cdot 9$

Step 2.3

Simplify each term.

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Step 2.3.1

Rewrite using the commutative property of multiplication.

$e^{4 x} \frac{d y}{d x} + 4 e^{4 x} y = 7 e^{4 x} x + e^{4 x} \cdot 9$

Step 2.3.2

Move $9$ to the left of $e^{4 x}$ .

$e^{4 x} \frac{d y}{d x} + 4 e^{4 x} y = 7 e^{4 x} x + 9 e^{4 x}$

Step 2.4

Reorder factors in $e^{4 x} \frac{d y}{d x} + 4 e^{4 x} y = 7 e^{4 x} x + 9 e^{4 x}$ .

$e^{4 x} \frac{d y}{d x} + 4 y e^{4 x} = 7 x e^{4 x} + 9 e^{4 x}$

Step 3

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

$\frac{d}{d x} [e^{4 x} y] = 7 x e^{4 x} + 9 e^{4 x}$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [e^{4 x} y] d x = \int 7 x e^{4 x} + 9 e^{4 x} d x$

Step 5

Integrate the left side.

$e^{4 x} y = \int 7 x e^{4 x} + 9 e^{4 x} d x$

Step 6

Integrate the right side.

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Step 6.1

Split the single integral into multiple integrals.

$e^{4 x} y = \int 7 x e^{4 x} d x + \int 9 e^{4 x} d x$

Step 6.2

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$e^{4 x} y = 7 \int x e^{4 x} d x + \int 9 e^{4 x} d x$

Step 6.3

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

$e^{4 x} y = 7 (x (\frac{1}{4} e^{4 x}) - \int \frac{1}{4} e^{4 x} d x) + \int 9 e^{4 x} d x$

Step 6.4

Simplify.

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Step 6.4.1

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

$e^{4 x} y = 7 (x \frac{e^{4 x}}{4} - \int \frac{1}{4} e^{4 x} d x) + \int 9 e^{4 x} d x$

Step 6.4.2

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

$e^{4 x} y = 7 (\frac{x e^{4 x}}{4} - \int \frac{1}{4} e^{4 x} d x) + \int 9 e^{4 x} d x$

Step 6.4.3

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

$e^{4 x} y = 7 (\frac{x e^{4 x}}{4} - \int \frac{e^{4 x}}{4} d x) + \int 9 e^{4 x} d x$

Step 6.5

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

$e^{4 x} y = 7 (\frac{x e^{4 x}}{4} - (\frac{1}{4} \int e^{4 x} d x)) + \int 9 e^{4 x} d x$

Step 6.6

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 6.6.1

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

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Step 6.6.1.1

Differentiate $4 x$ .

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

Step 6.6.1.2

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 6.6.1.3

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 6.6.1.4

Multiply $4$ by $1$ .

$4$

Step 6.6.2

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

$e^{4 x} y = 7 (\frac{x e^{4 x}}{4} - \frac{1}{4} \int e^{u_{1}} \frac{1}{4} d u_{1}) + \int 9 e^{4 x} d x$

Step 6.7

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

$e^{4 x} y = 7 (\frac{x e^{4 x}}{4} - \frac{1}{4} \int \frac{e^{u_{1}}}{4} d u_{1}) + \int 9 e^{4 x} d x$

Step 6.8

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

$e^{4 x} y = 7 (\frac{x e^{4 x}}{4} - \frac{1}{4} (\frac{1}{4} \int e^{u_{1}} d u_{1})) + \int 9 e^{4 x} d x$

Step 6.9

Simplify.

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Step 6.9.1

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

$e^{4 x} y = 7 (\frac{x e^{4 x}}{4} - \frac{1}{4 \cdot 4} \int e^{u_{1}} d u_{1}) + \int 9 e^{4 x} d x$

Step 6.9.2

Multiply $4$ by $4$ .

$e^{4 x} y = 7 (\frac{x e^{4 x}}{4} - \frac{1}{16} \int e^{u_{1}} d u_{1}) + \int 9 e^{4 x} d x$

Step 6.10

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

$e^{4 x} y = 7 (\frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C)) + \int 9 e^{4 x} d x$

Step 6.11

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$e^{4 x} y = 7 (\frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C)) + 9 \int e^{4 x} d x$

Step 6.12

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 6.12.1

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

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Step 6.12.1.1

Differentiate $4 x$ .

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

Step 6.12.1.2

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 6.12.1.3

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 6.12.1.4

Multiply $4$ by $1$ .

$4$

Step 6.12.2

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

$e^{4 x} y = 7 (\frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C)) + 9 \int e^{u_{2}} \frac{1}{4} d u_{2}$

Step 6.13

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

$e^{4 x} y = 7 (\frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C)) + 9 \int \frac{e^{u_{2}}}{4} d u_{2}$

Step 6.14

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

$e^{4 x} y = 7 (\frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C)) + 9 (\frac{1}{4} \int e^{u_{2}} d u_{2})$

Step 6.15

Combine $\frac{1}{4}$ and $9$ .

$e^{4 x} y = 7 (\frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C)) + \frac{9}{4} \int e^{u_{2}} d u_{2}$

Step 6.16

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

$e^{4 x} y = 7 (\frac{x e^{4 x}}{4} - \frac{1}{16} (e^{u_{1}} + C)) + \frac{9}{4} (e^{u_{2}} + C)$

Step 6.17

Simplify.

$e^{4 x} y = 7 (\frac{1}{4} x e^{4 x} - \frac{1}{16} e^{u_{1}}) + \frac{9}{4} e^{u_{2}} + C$

Step 6.18

Substitute back in for each integration substitution variable.

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Step 6.18.1

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

$e^{4 x} y = 7 (\frac{1}{4} x e^{4 x} - \frac{1}{16} e^{4 x}) + \frac{9}{4} e^{u_{2}} + C$

Step 6.18.2

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

$e^{4 x} y = 7 (\frac{1}{4} x e^{4 x} - \frac{1}{16} e^{4 x}) + \frac{9}{4} e^{4 x} + C$

Step 7

Solve for $y$ .

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Step 7.1

Simplify.

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Step 7.1.1

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

$e^{4 x} y = 7 (\frac{1}{4} \cdot (x e^{4 x}) - \frac{e^{4 x}}{16}) + \frac{9}{4} \cdot e^{4 x} + C$

Step 7.1.2

Remove parentheses.

$e^{4 x} y = 7 (\frac{1}{4} \cdot x e^{4 x} - \frac{e^{4 x}}{16}) + \frac{9}{4} \cdot e^{4 x} + C$

Step 7.2

Divide each term in $e^{4 x} y = 7 (\frac{1}{4} \cdot (x e^{4 x}) - \frac{e^{4 x}}{16}) + \frac{9}{4} \cdot e^{4 x} + C$ by $e^{4 x}$ and simplify.

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Step 7.2.1

Divide each term in $e^{4 x} y = 7 (\frac{1}{4} \cdot (x e^{4 x}) - \frac{e^{4 x}}{16}) + \frac{9}{4} \cdot e^{4 x} + C$ by $e^{4 x}$ .

$\frac{e^{4 x} y}{e^{4 x}} = \frac{7 (\frac{1}{4} \cdot (x e^{4 x}) - \frac{e^{4 x}}{16})}{e^{4 x}} + \frac{\frac{9}{4} \cdot e^{4 x}}{e^{4 x}} + \frac{C}{e^{4 x}}$

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.

$\frac{e^{4 x} y}{e^{4 x}} = \frac{7 (\frac{1}{4} \cdot (x e^{4 x}) - \frac{e^{4 x}}{16})}{e^{4 x}} + \frac{\frac{9}{4} \cdot e^{4 x}}{e^{4 x}} + \frac{C}{e^{4 x}}$

Step 7.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{7 (\frac{1}{4} \cdot (x e^{4 x}) - \frac{e^{4 x}}{16})}{e^{4 x}} + \frac{\frac{9}{4} \cdot e^{4 x}}{e^{4 x}} + \frac{C}{e^{4 x}}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Combine the numerators over the common denominator.

$y = \frac{7 (\frac{1}{4} \cdot (x e^{4 x}) - \frac{e^{4 x}}{16}) + \frac{9}{4} \cdot e^{4 x} + C}{e^{4 x}}$

Step 7.2.3.2

Simplify each term.

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Step 7.2.3.2.1

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

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Step 7.2.3.2.1.1

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

$y = \frac{7 (\frac{x}{4} e^{4 x} - \frac{e^{4 x}}{16}) + \frac{9}{4} \cdot e^{4 x} + C}{e^{4 x}}$

Step 7.2.3.2.1.2

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

$y = \frac{7 (\frac{x e^{4 x}}{4} - \frac{e^{4 x}}{16}) + \frac{9}{4} \cdot e^{4 x} + C}{e^{4 x}}$

Step 7.2.3.2.2

Apply the distributive property.

$y = \frac{7 \frac{x e^{4 x}}{4} + 7 (- \frac{e^{4 x}}{16}) + \frac{9}{4} \cdot e^{4 x} + C}{e^{4 x}}$

Step 7.2.3.2.3

Combine $7$ and $\frac{x e^{4 x}}{4}$ .

$y = \frac{\frac{7 (x e^{4 x})}{4} + 7 (- \frac{e^{4 x}}{16}) + \frac{9}{4} \cdot e^{4 x} + C}{e^{4 x}}$

Step 7.2.3.2.4

Multiply $7 (- \frac{e^{4 x}}{16})$ .

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Step 7.2.3.2.4.1

Multiply $- 1$ by $7$ .

$y = \frac{\frac{7 (x e^{4 x})}{4} - 7 \frac{e^{4 x}}{16} + \frac{9}{4} \cdot e^{4 x} + C}{e^{4 x}}$

Step 7.2.3.2.4.2

Combine $- 7$ and $\frac{e^{4 x}}{16}$ .

$y = \frac{\frac{7 (x e^{4 x})}{4} + \frac{- 7 e^{4 x}}{16} + \frac{9}{4} \cdot e^{4 x} + C}{e^{4 x}}$

Step 7.2.3.2.5

Move the negative in front of the fraction.

$y = \frac{\frac{7 x e^{4 x}}{4} - \frac{7 e^{4 x}}{16} + \frac{9}{4} \cdot e^{4 x} + C}{e^{4 x}}$

Step 7.2.3.2.6

Combine $\frac{9}{4}$ and $e^{4 x}$ .

$y = \frac{\frac{7 x e^{4 x}}{4} - \frac{7 e^{4 x}}{16} + \frac{9 e^{4 x}}{4} + C}{e^{4 x}}$

Step 7.2.3.3

To write $\frac{9 e^{4 x}}{4}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \frac{\frac{7 x e^{4 x}}{4} - \frac{7 e^{4 x}}{16} + \frac{9 e^{4 x}}{4} \cdot \frac{4}{4} + C}{e^{4 x}}$

Step 7.2.3.4

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.2.3.4.1

Multiply $\frac{9 e^{4 x}}{4}$ by $\frac{4}{4}$ .

$y = \frac{\frac{7 x e^{4 x}}{4} - \frac{7 e^{4 x}}{16} + \frac{9 e^{4 x} \cdot 4}{4 \cdot 4} + C}{e^{4 x}}$

Step 7.2.3.4.2

Multiply $4$ by $4$ .

$y = \frac{\frac{7 x e^{4 x}}{4} - \frac{7 e^{4 x}}{16} + \frac{9 e^{4 x} \cdot 4}{16} + C}{e^{4 x}}$

Step 7.2.3.5

Simplify terms.

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Step 7.2.3.5.1

Combine the numerators over the common denominator.

$y = \frac{\frac{7 x e^{4 x}}{4} + \frac{- 7 e^{4 x} + 9 e^{4 x} \cdot 4}{16} + C}{e^{4 x}}$

Step 7.2.3.5.2

Simplify each term.

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Step 7.2.3.5.2.1

Simplify the numerator.

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Step 7.2.3.5.2.1.1

Factor $e^{4 x}$ out of $- 7 e^{4 x} + 9 e^{4 x} \cdot 4$ .

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Step 7.2.3.5.2.1.1.1

Factor $e^{4 x}$ out of $- 7 e^{4 x}$ .

$y = \frac{\frac{7 x e^{4 x}}{4} + \frac{e^{4 x} \cdot - 7 + 9 e^{4 x} \cdot 4}{16} + C}{e^{4 x}}$

Step 7.2.3.5.2.1.1.2

Factor $e^{4 x}$ out of $9 e^{4 x} \cdot 4$ .

$y = \frac{\frac{7 x e^{4 x}}{4} + \frac{e^{4 x} \cdot - 7 + e^{4 x} (9 \cdot 4)}{16} + C}{e^{4 x}}$

Step 7.2.3.5.2.1.1.3

Factor $e^{4 x}$ out of $e^{4 x} \cdot - 7 + e^{4 x} (9 \cdot 4)$ .

$y = \frac{\frac{7 x e^{4 x}}{4} + \frac{e^{4 x} (- 7 + 9 \cdot 4)}{16} + C}{e^{4 x}}$

Step 7.2.3.5.2.1.2

Multiply $9$ by $4$ .

$y = \frac{\frac{7 x e^{4 x}}{4} + \frac{e^{4 x} (- 7 + 36)}{16} + C}{e^{4 x}}$

Step 7.2.3.5.2.1.3

Add $- 7$ and $36$ .

$y = \frac{\frac{7 x e^{4 x}}{4} + \frac{e^{4 x} \cdot 29}{16} + C}{e^{4 x}}$

Step 7.2.3.5.2.2

Move $29$ to the left of $e^{4 x}$ .

$y = \frac{\frac{7 x e^{4 x}}{4} + \frac{29 e^{4 x}}{16} + C}{e^{4 x}}$

Step 7.2.3.6

Simplify the numerator.

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Step 7.2.3.6.1

To write $\frac{7 x e^{4 x}}{4}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \frac{\frac{7 x e^{4 x}}{4} \cdot \frac{4}{4} + \frac{29 e^{4 x}}{16} + C}{e^{4 x}}$

Step 7.2.3.6.2

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.2.3.6.2.1

Multiply $\frac{7 x e^{4 x}}{4}$ by $\frac{4}{4}$ .

$y = \frac{\frac{7 x e^{4 x} \cdot 4}{4 \cdot 4} + \frac{29 e^{4 x}}{16} + C}{e^{4 x}}$

Step 7.2.3.6.2.2

Multiply $4$ by $4$ .

$y = \frac{\frac{7 x e^{4 x} \cdot 4}{16} + \frac{29 e^{4 x}}{16} + C}{e^{4 x}}$

Step 7.2.3.6.3

Combine the numerators over the common denominator.

$y = \frac{\frac{7 x e^{4 x} \cdot 4 + 29 e^{4 x}}{16} + C}{e^{4 x}}$

Step 7.2.3.6.4

Simplify the numerator.

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Step 7.2.3.6.4.1

Factor $e^{4 x}$ out of $7 x e^{4 x} \cdot 4 + 29 e^{4 x}$ .

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Step 7.2.3.6.4.1.1

Factor $e^{4 x}$ out of $7 x e^{4 x} \cdot 4$ .

$y = \frac{\frac{e^{4 x} (7 x \cdot 4) + 29 e^{4 x}}{16} + C}{e^{4 x}}$

Step 7.2.3.6.4.1.2

Factor $e^{4 x}$ out of $29 e^{4 x}$ .

$y = \frac{\frac{e^{4 x} (7 x \cdot 4) + e^{4 x} \cdot 29}{16} + C}{e^{4 x}}$

Step 7.2.3.6.4.1.3

Factor $e^{4 x}$ out of $e^{4 x} (7 x \cdot 4) + e^{4 x} \cdot 29$ .

$y = \frac{\frac{e^{4 x} (7 x \cdot 4 + 29)}{16} + C}{e^{4 x}}$

Step 7.2.3.6.4.2

Multiply $4$ by $7$ .

$y = \frac{\frac{e^{4 x} (28 x + 29)}{16} + C}{e^{4 x}}$

Step 7.2.3.6.5

To write $C$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$y = \frac{\frac{e^{4 x} (28 x + 29)}{16} + C \cdot \frac{16}{16}}{e^{4 x}}$

Step 7.2.3.6.6

Combine $C$ and $\frac{16}{16}$ .

$y = \frac{\frac{e^{4 x} (28 x + 29)}{16} + \frac{C \cdot 16}{16}}{e^{4 x}}$

Step 7.2.3.6.7

Combine the numerators over the common denominator.

$y = \frac{\frac{e^{4 x} (28 x + 29) + C \cdot 16}{16}}{e^{4 x}}$

Step 7.2.3.6.8

Simplify the numerator.

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Step 7.2.3.6.8.1

Apply the distributive property.

$y = \frac{\frac{e^{4 x} (28 x) + e^{4 x} \cdot 29 + C \cdot 16}{16}}{e^{4 x}}$

Step 7.2.3.6.8.2

Rewrite using the commutative property of multiplication.

$y = \frac{\frac{28 e^{4 x} x + e^{4 x} \cdot 29 + C \cdot 16}{16}}{e^{4 x}}$

Step 7.2.3.6.8.3

Move $29$ to the left of $e^{4 x}$ .

$y = \frac{\frac{28 e^{4 x} x + 29 \cdot e^{4 x} + C \cdot 16}{16}}{e^{4 x}}$

Step 7.2.3.6.8.4

Move $16$ to the left of $C$ .

$y = \frac{\frac{28 e^{4 x} x + 29 e^{4 x} + 16 C}{16}}{e^{4 x}}$

Step 7.2.3.7

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{28 e^{4 x} x + 29 e^{4 x} + 16 C}{16} \cdot \frac{1}{e^{4 x}}$

Step 7.2.3.8

Multiply $\frac{28 e^{4 x} x + 29 e^{4 x} + 16 C}{16}$ by $\frac{1}{e^{4 x}}$ .

$y = \frac{28 e^{4 x} x + 29 e^{4 x} + 16 C}{16 e^{4 x}}$

Step 7.2.3.9

Reorder factors in $\frac{28 e^{4 x} x + 29 e^{4 x} + 16 C}{16 e^{4 x}}$ .

$y = \frac{28 x e^{4 x} + 29 e^{4 x} + 16 C}{16 e^{4 x}}$