Calculus Examples

$\frac{d y}{d x} = 1 - x + 4 y$

Step 1

Rewrite the equation as $M (x) \frac{d y}{d x} + P (x) y = Q (x)$ .

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Step 1.1

Subtract $4 y$ from both sides of the equation.

$\frac{d y}{d x} - 4 y = 1 - x$

Step 1.2

Reorder terms.

$\frac{d y}{d x} - 4 y = - x + 1$

Step 2

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

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Step 2.1

Set up the integration.

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

Step 2.2

Apply the constant rule.

$e^{- 4 x + C}$

Step 2.3

Remove the constant of integration.

$e^{- 4 x}$

Step 3

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

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Step 3.1

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

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

Simplify each term.

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Step 3.3.1

Rewrite using the commutative property of multiplication.

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

Step 3.3.2

Multiply $e^{- 4 x}$ by $1$ .

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

Step 3.4

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

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

Step 4

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

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

Step 5

Set up an integral on each side.

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

Step 6

Integrate the left side.

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

Step 7

Integrate the right side.

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Step 7.1

Split the single integral into multiple integrals.

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

Step 7.2

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 7.3

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

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

Step 7.4

Simplify.

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Step 7.4.1

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

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

Step 7.4.2

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

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

Step 7.4.3

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

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

Step 7.5

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 7.6

Simplify.

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Step 7.6.1

Multiply $- 1$ by $- 1$ .

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

Step 7.6.2

Multiply $\int \frac{e^{- 4 x}}{4} d x$ by $1$ .

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

Step 7.7

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

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

Step 7.8

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 7.8.1

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

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Step 7.8.1.1

Differentiate $- 4 x$ .

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

Step 7.8.1.2

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 7.8.1.3

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 7.8.1.4

Multiply $- 4$ by $1$ .

$- 4$

Step 7.8.2

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

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

Step 7.9

Simplify.

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Step 7.9.1

Move the negative in front of the fraction.

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

Step 7.9.2

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

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

Step 7.10

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

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

Step 7.11

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

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

Step 7.12

Simplify.

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Step 7.12.1

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

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

Step 7.12.2

Multiply $4$ by $4$ .

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

Step 7.13

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

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

Step 7.14

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 7.14.1

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

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Step 7.14.1.1

Differentiate $- 4 x$ .

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

Step 7.14.1.2

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 7.14.1.3

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 7.14.1.4

Multiply $- 4$ by $1$ .

$- 4$

Step 7.14.2

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

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

Step 7.15

Simplify.

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Step 7.15.1

Move the negative in front of the fraction.

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

Step 7.15.2

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

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

Step 7.16

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

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

Step 7.17

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

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

Step 7.18

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

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

Step 7.19

Simplify.

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Step 7.19.1

Simplify.

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

Step 7.19.2

Simplify.

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Step 7.19.2.1

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

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

Step 7.19.2.2

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

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

Step 7.19.2.3

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

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

Step 7.20

Substitute back in for each integration substitution variable.

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Step 7.20.1

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

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

Step 7.20.2

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

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

Step 7.21

Simplify.

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Step 7.21.1

Apply the distributive property.

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

Step 7.21.2

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

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Step 7.21.2.1

Multiply $- 1$ by $- 1$ .

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

Step 7.21.2.2

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

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

Step 7.21.3

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

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Step 7.21.3.1

Multiply $- 1$ by $- 1$ .

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

Step 7.21.3.2

Multiply $\frac{e^{- 4 x}}{16}$ by $1$ .

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

Step 7.21.4

Reorder factors in $\frac{e^{- 4 x} x}{4} + \frac{e^{- 4 x}}{16}$ .

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

Step 7.21.5

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

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

Step 7.22

Reorder terms.

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

Step 8

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

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Step 8.1

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

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

Step 8.2

Simplify the left side.

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Step 8.2.1

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

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $y$ by $1$ .

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

Step 8.3

Simplify the right side.

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Step 8.3.1

Combine the numerators over the common denominator.

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

Step 8.3.2

Simplify each term.

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Step 8.3.2.1

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

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

Step 8.3.2.2

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

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

Step 8.3.2.3

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

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

Step 8.3.2.4

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

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

Step 8.3.3

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

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

Step 8.3.4

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

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Step 8.3.4.1

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

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

Step 8.3.4.2

Multiply $4$ by $4$ .

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

Step 8.3.5

Simplify terms.

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Step 8.3.5.1

Combine the numerators over the common denominator.

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

Step 8.3.5.2

Simplify each term.

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Step 8.3.5.2.1

Simplify the numerator.

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Step 8.3.5.2.1.1

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

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Step 8.3.5.2.1.1.1

Multiply by $1$ .

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

Step 8.3.5.2.1.1.2

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

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

Step 8.3.5.2.1.1.3

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

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

Step 8.3.5.2.1.2

Multiply $- 1$ by $4$ .

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

Step 8.3.5.2.1.3

Subtract $4$ from $1$ .

$y = \frac{\frac{x e^{- 4 x}}{4} + \frac{e^{- 4 x} \cdot - 3}{16} + C}{e^{- 4 x}}$

Step 8.3.5.2.2

Move $- 3$ to the left of $e^{- 4 x}$ .

$y = \frac{\frac{x e^{- 4 x}}{4} + \frac{- 3 \cdot e^{- 4 x}}{16} + C}{e^{- 4 x}}$

Step 8.3.5.2.3

Move the negative in front of the fraction.

$y = \frac{\frac{x e^{- 4 x}}{4} - \frac{3 e^{- 4 x}}{16} + C}{e^{- 4 x}}$

Step 8.3.6

Simplify the numerator.

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Step 8.3.6.1

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

$y = \frac{\frac{x e^{- 4 x}}{4} \cdot \frac{4}{4} - \frac{3 e^{- 4 x}}{16} + C}{e^{- 4 x}}$

Step 8.3.6.2

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

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Step 8.3.6.2.1

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

$y = \frac{\frac{x e^{- 4 x} \cdot 4}{4 \cdot 4} - \frac{3 e^{- 4 x}}{16} + C}{e^{- 4 x}}$

Step 8.3.6.2.2

Multiply $4$ by $4$ .

$y = \frac{\frac{x e^{- 4 x} \cdot 4}{16} - \frac{3 e^{- 4 x}}{16} + C}{e^{- 4 x}}$

Step 8.3.6.3

Combine the numerators over the common denominator.

$y = \frac{\frac{x e^{- 4 x} \cdot 4 - 3 e^{- 4 x}}{16} + C}{e^{- 4 x}}$

Step 8.3.6.4

Simplify the numerator.

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Step 8.3.6.4.1

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

$y = \frac{\frac{4 \cdot (x e^{- 4 x}) - 3 e^{- 4 x}}{16} + C}{e^{- 4 x}}$

Step 8.3.6.4.2

Factor $e^{- 4 x}$ out of $4 x e^{- 4 x} - 3 e^{- 4 x}$ .

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Step 8.3.6.4.2.1

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

$y = \frac{\frac{e^{- 4 x} (4 x) - 3 e^{- 4 x}}{16} + C}{e^{- 4 x}}$

Step 8.3.6.4.2.2

Factor $e^{- 4 x}$ out of $- 3 e^{- 4 x}$ .

$y = \frac{\frac{e^{- 4 x} (4 x) + e^{- 4 x} \cdot - 3}{16} + C}{e^{- 4 x}}$

Step 8.3.6.4.2.3

Factor $e^{- 4 x}$ out of $e^{- 4 x} (4 x) + e^{- 4 x} \cdot - 3$ .

$y = \frac{\frac{e^{- 4 x} (4 x - 3)}{16} + C}{e^{- 4 x}}$

Step 8.3.6.5

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

$y = \frac{\frac{e^{- 4 x} (4 x - 3)}{16} + C \cdot \frac{16}{16}}{e^{- 4 x}}$

Step 8.3.6.6

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

$y = \frac{\frac{e^{- 4 x} (4 x - 3)}{16} + \frac{C \cdot 16}{16}}{e^{- 4 x}}$

Step 8.3.6.7

Combine the numerators over the common denominator.

$y = \frac{\frac{e^{- 4 x} (4 x - 3) + C \cdot 16}{16}}{e^{- 4 x}}$

Step 8.3.6.8

Simplify the numerator.

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Step 8.3.6.8.1

Apply the distributive property.

$y = \frac{\frac{e^{- 4 x} (4 x) + e^{- 4 x} \cdot - 3 + C \cdot 16}{16}}{e^{- 4 x}}$

Step 8.3.6.8.2

Rewrite using the commutative property of multiplication.

$y = \frac{\frac{4 e^{- 4 x} x + e^{- 4 x} \cdot - 3 + C \cdot 16}{16}}{e^{- 4 x}}$

Step 8.3.6.8.3

Move $- 3$ to the left of $e^{- 4 x}$ .

$y = \frac{\frac{4 e^{- 4 x} x - 3 \cdot e^{- 4 x} + C \cdot 16}{16}}{e^{- 4 x}}$

Step 8.3.6.8.4

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

$y = \frac{\frac{4 e^{- 4 x} x - 3 e^{- 4 x} + 16 C}{16}}{e^{- 4 x}}$

Step 8.3.7

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{4 e^{- 4 x} x - 3 e^{- 4 x} + 16 C}{16} \cdot \frac{1}{e^{- 4 x}}$

Step 8.3.8

Multiply $\frac{4 e^{- 4 x} x - 3 e^{- 4 x} + 16 C}{16}$ by $\frac{1}{e^{- 4 x}}$ .

$y = \frac{4 e^{- 4 x} x - 3 e^{- 4 x} + 16 C}{16 e^{- 4 x}}$

Step 8.3.9

Reorder factors in $\frac{4 e^{- 4 x} x - 3 e^{- 4 x} + 16 C}{16 e^{- 4 x}}$ .

$y = \frac{4 x e^{- 4 x} - 3 e^{- 4 x} + 16 C}{16 e^{- 4 x}}$