Calculus Examples

$\frac{d y}{d x} + 3 y = 6 x + 11$

Step 1

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = 3$ .

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Step 1.1

Set up the integration.

$e^{\int 3 d x}$

Step 1.2

Apply the constant rule.

$e^{3 x + C}$

Step 1.3

Remove the constant of integration.

$e^{3 x}$

Step 2

Multiply each term by the integrating factor $e^{3 x}$ .

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Step 2.1

Multiply each term by $e^{3 x}$ .

$e^{3 x} \frac{d y}{d x} + e^{3 x} (3 y) = e^{3 x} (6 x) + e^{3 x} \cdot 11$

Step 2.2

Rewrite using the commutative property of multiplication.

$e^{3 x} \frac{d y}{d x} + 3 e^{3 x} y = e^{3 x} (6 x) + e^{3 x} \cdot 11$

Step 2.3

Simplify each term.

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Step 2.3.1

Rewrite using the commutative property of multiplication.

$e^{3 x} \frac{d y}{d x} + 3 e^{3 x} y = 6 e^{3 x} x + e^{3 x} \cdot 11$

Step 2.3.2

Move $11$ to the left of $e^{3 x}$ .

$e^{3 x} \frac{d y}{d x} + 3 e^{3 x} y = 6 e^{3 x} x + 11 e^{3 x}$

Step 2.4

Reorder factors in $e^{3 x} \frac{d y}{d x} + 3 e^{3 x} y = 6 e^{3 x} x + 11 e^{3 x}$ .

$e^{3 x} \frac{d y}{d x} + 3 y e^{3 x} = 6 x e^{3 x} + 11 e^{3 x}$

Step 3

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

$\frac{d}{d x} [e^{3 x} y] = 6 x e^{3 x} + 11 e^{3 x}$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [e^{3 x} y] d x = \int 6 x e^{3 x} + 11 e^{3 x} d x$

Step 5

Integrate the left side.

$e^{3 x} y = \int 6 x e^{3 x} + 11 e^{3 x} d x$

Step 6

Integrate the right side.

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Step 6.1

Split the single integral into multiple integrals.

$e^{3 x} y = \int 6 x e^{3 x} d x + \int 11 e^{3 x} d x$

Step 6.2

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$e^{3 x} y = 6 \int x e^{3 x} d x + \int 11 e^{3 x} d x$

Step 6.3

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{3 x}$ .

$e^{3 x} y = 6 (x (\frac{1}{3} e^{3 x}) - \int \frac{1}{3} e^{3 x} d x) + \int 11 e^{3 x} d x$

Step 6.4

Simplify.

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Step 6.4.1

Combine $\frac{1}{3}$ and $e^{3 x}$ .

$e^{3 x} y = 6 (x \frac{e^{3 x}}{3} - \int \frac{1}{3} e^{3 x} d x) + \int 11 e^{3 x} d x$

Step 6.4.2

Combine $x$ and $\frac{e^{3 x}}{3}$ .

$e^{3 x} y = 6 (\frac{x e^{3 x}}{3} - \int \frac{1}{3} e^{3 x} d x) + \int 11 e^{3 x} d x$

Step 6.4.3

Combine $\frac{1}{3}$ and $e^{3 x}$ .

$e^{3 x} y = 6 (\frac{x e^{3 x}}{3} - \int \frac{e^{3 x}}{3} d x) + \int 11 e^{3 x} d x$

Step 6.5

Since $\frac{1}{3}$ is constant with respect to $x$ , move $\frac{1}{3}$ out of the integral.

$e^{3 x} y = 6 (\frac{x e^{3 x}}{3} - (\frac{1}{3} \int e^{3 x} d x)) + \int 11 e^{3 x} d x$

Step 6.6

Let $u_{1} = 3 x$ . Then $d u_{1} = 3 d x$ , so $\frac{1}{3} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 6.6.1

Let $u_{1} = 3 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 6.6.1.1

Differentiate $3 x$ .

$\frac{d}{d x} [3 x]$

Step 6.6.1.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x]$

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 $n = 1$ .

$3 \cdot 1$

Step 6.6.1.4

Multiply $3$ by $1$ .

$3$

Step 6.6.2

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

$e^{3 x} y = 6 (\frac{x e^{3 x}}{3} - \frac{1}{3} \int e^{u_{1}} \frac{1}{3} d u_{1}) + \int 11 e^{3 x} d x$

Step 6.7

Combine $e^{u_{1}}$ and $\frac{1}{3}$ .

$e^{3 x} y = 6 (\frac{x e^{3 x}}{3} - \frac{1}{3} \int \frac{e^{u_{1}}}{3} d u_{1}) + \int 11 e^{3 x} d x$

Step 6.8

Since $\frac{1}{3}$ is constant with respect to $u_{1}$ , move $\frac{1}{3}$ out of the integral.

$e^{3 x} y = 6 (\frac{x e^{3 x}}{3} - \frac{1}{3} (\frac{1}{3} \int e^{u_{1}} d u_{1})) + \int 11 e^{3 x} d x$

Step 6.9

Simplify.

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Step 6.9.1

Multiply $\frac{1}{3}$ by $\frac{1}{3}$ .

$e^{3 x} y = 6 (\frac{x e^{3 x}}{3} - \frac{1}{3 \cdot 3} \int e^{u_{1}} d u_{1}) + \int 11 e^{3 x} d x$

Step 6.9.2

Multiply $3$ by $3$ .

$e^{3 x} y = 6 (\frac{x e^{3 x}}{3} - \frac{1}{9} \int e^{u_{1}} d u_{1}) + \int 11 e^{3 x} d x$

Step 6.10

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

$e^{3 x} y = 6 (\frac{x e^{3 x}}{3} - \frac{1}{9} (e^{u_{1}} + C)) + \int 11 e^{3 x} d x$

Step 6.11

Since $11$ is constant with respect to $x$ , move $11$ out of the integral.

$e^{3 x} y = 6 (\frac{x e^{3 x}}{3} - \frac{1}{9} (e^{u_{1}} + C)) + 11 \int e^{3 x} d x$

Step 6.12

Let $u_{2} = 3 x$ . Then $d u_{2} = 3 d x$ , so $\frac{1}{3} d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 6.12.1

Let $u_{2} = 3 x$ . Find $\frac{d u_{2}}{d x}$ .

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Step 6.12.1.1

Differentiate $3 x$ .

$\frac{d}{d x} [3 x]$

Step 6.12.1.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x]$

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 $n = 1$ .

$3 \cdot 1$

Step 6.12.1.4

Multiply $3$ by $1$ .

$3$

Step 6.12.2

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

$e^{3 x} y = 6 (\frac{x e^{3 x}}{3} - \frac{1}{9} (e^{u_{1}} + C)) + 11 \int e^{u_{2}} \frac{1}{3} d u_{2}$

Step 6.13

Combine $e^{u_{2}}$ and $\frac{1}{3}$ .

$e^{3 x} y = 6 (\frac{x e^{3 x}}{3} - \frac{1}{9} (e^{u_{1}} + C)) + 11 \int \frac{e^{u_{2}}}{3} d u_{2}$

Step 6.14

Since $\frac{1}{3}$ is constant with respect to $u_{2}$ , move $\frac{1}{3}$ out of the integral.

$e^{3 x} y = 6 (\frac{x e^{3 x}}{3} - \frac{1}{9} (e^{u_{1}} + C)) + 11 (\frac{1}{3} \int e^{u_{2}} d u_{2})$

Step 6.15

Combine $\frac{1}{3}$ and $11$ .

$e^{3 x} y = 6 (\frac{x e^{3 x}}{3} - \frac{1}{9} (e^{u_{1}} + C)) + \frac{11}{3} \int e^{u_{2}} d u_{2}$

Step 6.16

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

$e^{3 x} y = 6 (\frac{x e^{3 x}}{3} - \frac{1}{9} (e^{u_{1}} + C)) + \frac{11}{3} (e^{u_{2}} + C)$

Step 6.17

Simplify.

$e^{3 x} y = 6 (\frac{1}{3} x e^{3 x} - \frac{1}{9} e^{u_{1}}) + \frac{11}{3} e^{u_{2}} + C$

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 $3 x$ .

$e^{3 x} y = 6 (\frac{1}{3} x e^{3 x} - \frac{1}{9} e^{3 x}) + \frac{11}{3} e^{u_{2}} + C$

Step 6.18.2

Replace all occurrences of $u_{2}$ with $3 x$ .

$e^{3 x} y = 6 (\frac{1}{3} x e^{3 x} - \frac{1}{9} e^{3 x}) + \frac{11}{3} e^{3 x} + C$

Step 6.19

Simplify.

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Step 6.19.1

Simplify each term.

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Step 6.19.1.1

Combine $\frac{1}{3}$ and $x$ .

$e^{3 x} y = 6 (\frac{x}{3} e^{3 x} - \frac{1}{9} e^{3 x}) + \frac{11}{3} e^{3 x} + C$

Step 6.19.1.2

Combine $\frac{x}{3}$ and $e^{3 x}$ .

$e^{3 x} y = 6 (\frac{x e^{3 x}}{3} - \frac{1}{9} e^{3 x}) + \frac{11}{3} e^{3 x} + C$

Step 6.19.1.3

Combine $e^{3 x}$ and $\frac{1}{9}$ .

$e^{3 x} y = 6 (\frac{x e^{3 x}}{3} - \frac{e^{3 x}}{9}) + \frac{11}{3} e^{3 x} + C$

Step 6.19.2

Apply the distributive property.

$e^{3 x} y = 6 \frac{x e^{3 x}}{3} + 6 (- \frac{e^{3 x}}{9}) + \frac{11}{3} e^{3 x} + C$

Step 6.19.3

Cancel the common factor of $3$ .

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Step 6.19.3.1

Factor $3$ out of $6$ .

$e^{3 x} y = 3 (2) \frac{x e^{3 x}}{3} + 6 (- \frac{e^{3 x}}{9}) + \frac{11}{3} e^{3 x} + C$

Step 6.19.3.2

Cancel the common factor.

$e^{3 x} y = 3 \cdot 2 \frac{x e^{3 x}}{3} + 6 (- \frac{e^{3 x}}{9}) + \frac{11}{3} e^{3 x} + C$

Step 6.19.3.3

Rewrite the expression.

$e^{3 x} y = 2 (x e^{3 x}) + 6 (- \frac{e^{3 x}}{9}) + \frac{11}{3} e^{3 x} + C$

Step 6.19.4

Cancel the common factor of $3$ .

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Step 6.19.4.1

Move the leading negative in $- \frac{e^{3 x}}{9}$ into the numerator.

$e^{3 x} y = 2 (x e^{3 x}) + 6 \frac{- e^{3 x}}{9} + \frac{11}{3} e^{3 x} + C$

Step 6.19.4.2

Factor $3$ out of $6$ .

$e^{3 x} y = 2 (x e^{3 x}) + 3 (2) \frac{- e^{3 x}}{9} + \frac{11}{3} e^{3 x} + C$

Step 6.19.4.3

Factor $3$ out of $9$ .

$e^{3 x} y = 2 (x e^{3 x}) + 3 \cdot 2 \frac{- e^{3 x}}{3 \cdot 3} + \frac{11}{3} e^{3 x} + C$

Step 6.19.4.4

Cancel the common factor.

$e^{3 x} y = 2 (x e^{3 x}) + 3 \cdot 2 \frac{- e^{3 x}}{3 \cdot 3} + \frac{11}{3} e^{3 x} + C$

Step 6.19.4.5

Rewrite the expression.

$e^{3 x} y = 2 (x e^{3 x}) + 2 \frac{- e^{3 x}}{3} + \frac{11}{3} e^{3 x} + C$

Step 6.19.5

Combine $2$ and $\frac{- e^{3 x}}{3}$ .

$e^{3 x} y = 2 (x e^{3 x}) + \frac{2 (- e^{3 x})}{3} + \frac{11}{3} e^{3 x} + C$

Step 6.19.6

Multiply $- 1$ by $2$ .

$e^{3 x} y = 2 (x e^{3 x}) + \frac{- 2 e^{3 x}}{3} + \frac{11}{3} e^{3 x} + C$

Step 6.19.7

Move the negative in front of the fraction.

$e^{3 x} y = 2 x e^{3 x} - \frac{2 e^{3 x}}{3} + \frac{11}{3} e^{3 x} + C$

Step 6.19.8

To write $2 x e^{3 x}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$e^{3 x} y = 2 x e^{3 x} \cdot \frac{3}{3} - \frac{2 e^{3 x}}{3} + \frac{11}{3} e^{3 x} + C$

Step 6.19.9

Combine $2 x e^{3 x}$ and $\frac{3}{3}$ .

$e^{3 x} y = \frac{2 x e^{3 x} \cdot 3}{3} - \frac{2 e^{3 x}}{3} + \frac{11}{3} e^{3 x} + C$

Step 6.19.10

Combine the numerators over the common denominator.

$e^{3 x} y = \frac{2 x e^{3 x} \cdot 3 - 2 e^{3 x}}{3} + \frac{11}{3} e^{3 x} + C$

Step 6.19.11

Simplify the numerator.

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Step 6.19.11.1

Factor $2 e^{3 x}$ out of $2 x e^{3 x} \cdot 3 - 2 e^{3 x}$ .

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Step 6.19.11.1.1

Factor $2 e^{3 x}$ out of $2 x e^{3 x} \cdot 3$ .

$e^{3 x} y = \frac{2 e^{3 x} (x \cdot 3) - 2 e^{3 x}}{3} + \frac{11}{3} e^{3 x} + C$

Step 6.19.11.1.2

Factor $2 e^{3 x}$ out of $- 2 e^{3 x}$ .

$e^{3 x} y = \frac{2 e^{3 x} (x \cdot 3) + 2 e^{3 x} (- 1)}{3} + \frac{11}{3} e^{3 x} + C$

Step 6.19.11.1.3

Factor $2 e^{3 x}$ out of $2 e^{3 x} (x \cdot 3) + 2 e^{3 x} (- 1)$ .

$e^{3 x} y = \frac{2 e^{3 x} (x \cdot 3 - 1)}{3} + \frac{11}{3} e^{3 x} + C$

Step 6.19.11.2

Move $3$ to the left of $x$ .

$e^{3 x} y = \frac{2 e^{3 x} (3 x - 1)}{3} + \frac{11}{3} e^{3 x} + C$

$e^{3 x} y = \frac{2}{3} e^{3 x} (3 x - 1) + \frac{11}{3} e^{3 x} + C$

Step 7

Divide each term in $e^{3 x} y = \frac{2}{3} \cdot (e^{3 x} (3 x - 1)) + \frac{11}{3} \cdot e^{3 x} + C$ by $e^{3 x}$ and simplify.

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Step 7.1

Divide each term in $e^{3 x} y = \frac{2}{3} \cdot (e^{3 x} (3 x - 1)) + \frac{11}{3} \cdot e^{3 x} + C$ by $e^{3 x}$ .

$\frac{e^{3 x} y}{e^{3 x}} = \frac{\frac{2}{3} \cdot (e^{3 x} (3 x - 1))}{e^{3 x}} + \frac{\frac{11}{3} \cdot e^{3 x}}{e^{3 x}} + \frac{C}{e^{3 x}}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $e^{3 x}$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{e^{3 x} y}{e^{3 x}} = \frac{\frac{2}{3} \cdot (e^{3 x} (3 x - 1))}{e^{3 x}} + \frac{\frac{11}{3} \cdot e^{3 x}}{e^{3 x}} + \frac{C}{e^{3 x}}$

Step 7.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{2}{3} \cdot (e^{3 x} (3 x - 1))}{e^{3 x}} + \frac{\frac{11}{3} \cdot e^{3 x}}{e^{3 x}} + \frac{C}{e^{3 x}}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Combine the numerators over the common denominator.

$y = \frac{\frac{2}{3} \cdot (e^{3 x} (3 x - 1)) + \frac{11}{3} \cdot e^{3 x} + C}{e^{3 x}}$

Step 7.3.2

Simplify each term.

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Step 7.3.2.1

Apply the distributive property.

$y = \frac{\frac{2}{3} \cdot (e^{3 x} (3 x) + e^{3 x} \cdot - 1) + \frac{11}{3} \cdot e^{3 x} + C}{e^{3 x}}$

Step 7.3.2.2

Rewrite using the commutative property of multiplication.

$y = \frac{\frac{2}{3} \cdot (3 e^{3 x} x + e^{3 x} \cdot - 1) + \frac{11}{3} \cdot e^{3 x} + C}{e^{3 x}}$

Step 7.3.2.3

Move $- 1$ to the left of $e^{3 x}$ .

$y = \frac{\frac{2}{3} \cdot (3 e^{3 x} x - 1 \cdot e^{3 x}) + \frac{11}{3} \cdot e^{3 x} + C}{e^{3 x}}$

Step 7.3.2.4

Rewrite $- 1 e^{3 x}$ as $- e^{3 x}$ .

$y = \frac{\frac{2}{3} \cdot (3 e^{3 x} x - e^{3 x}) + \frac{11}{3} \cdot e^{3 x} + C}{e^{3 x}}$

Step 7.3.2.5

Apply the distributive property.

$y = \frac{\frac{2}{3} (3 e^{3 x} x) + \frac{2}{3} (- e^{3 x}) + \frac{11}{3} \cdot e^{3 x} + C}{e^{3 x}}$

Step 7.3.2.6

Cancel the common factor of $3$ .

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Step 7.3.2.6.1

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

$y = \frac{\frac{2}{3} (3 (e^{3 x} x)) + \frac{2}{3} (- e^{3 x}) + \frac{11}{3} \cdot e^{3 x} + C}{e^{3 x}}$

Step 7.3.2.6.2

Cancel the common factor.

$y = \frac{\frac{2}{3} (3 (e^{3 x} x)) + \frac{2}{3} (- e^{3 x}) + \frac{11}{3} \cdot e^{3 x} + C}{e^{3 x}}$

Step 7.3.2.6.3

Rewrite the expression.

$y = \frac{2 (e^{3 x} x) + \frac{2}{3} (- e^{3 x}) + \frac{11}{3} \cdot e^{3 x} + C}{e^{3 x}}$

Step 7.3.2.7

Combine $\frac{2}{3}$ and $e^{3 x}$ .

$y = \frac{2 (e^{3 x} x) - \frac{2 e^{3 x}}{3} + \frac{11}{3} \cdot e^{3 x} + C}{e^{3 x}}$

Step 7.3.2.8

Combine $2 e^{3 x} x$ and $- \frac{2 e^{3 x}}{3}$ using a common denominator.

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Step 7.3.2.8.1

Move $e^{3 x}$ .

$y = \frac{2 x e^{3 x} - \frac{2 e^{3 x}}{3} + \frac{11}{3} \cdot e^{3 x} + C}{e^{3 x}}$

Step 7.3.2.8.2

To write $2 x e^{3 x}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \frac{2 x e^{3 x} \cdot \frac{3}{3} - \frac{2 e^{3 x}}{3} + \frac{11}{3} \cdot e^{3 x} + C}{e^{3 x}}$

Step 7.3.2.8.3

Combine $2 x e^{3 x}$ and $\frac{3}{3}$ .

$y = \frac{\frac{2 x e^{3 x} \cdot 3}{3} - \frac{2 e^{3 x}}{3} + \frac{11}{3} \cdot e^{3 x} + C}{e^{3 x}}$

Step 7.3.2.8.4

Combine the numerators over the common denominator.

$y = \frac{\frac{2 x e^{3 x} \cdot 3 - 2 e^{3 x}}{3} + \frac{11}{3} \cdot e^{3 x} + C}{e^{3 x}}$

Step 7.3.2.9

Simplify the numerator.

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Step 7.3.2.9.1

Factor $2 e^{3 x}$ out of $2 x e^{3 x} \cdot 3 - 2 e^{3 x}$ .

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Step 7.3.2.9.1.1

Factor $2 e^{3 x}$ out of $2 x e^{3 x} \cdot 3$ .

$y = \frac{\frac{2 e^{3 x} (x \cdot 3) - 2 e^{3 x}}{3} + \frac{11}{3} \cdot e^{3 x} + C}{e^{3 x}}$

Step 7.3.2.9.1.2

Factor $2 e^{3 x}$ out of $- 2 e^{3 x}$ .

$y = \frac{\frac{2 e^{3 x} (x \cdot 3) + 2 e^{3 x} (- 1)}{3} + \frac{11}{3} \cdot e^{3 x} + C}{e^{3 x}}$

Step 7.3.2.9.1.3

Factor $2 e^{3 x}$ out of $2 e^{3 x} (x \cdot 3) + 2 e^{3 x} (- 1)$ .

$y = \frac{\frac{2 e^{3 x} (x \cdot 3 - 1)}{3} + \frac{11}{3} \cdot e^{3 x} + C}{e^{3 x}}$

Step 7.3.2.9.2

Move $3$ to the left of $x$ .

$y = \frac{\frac{2 e^{3 x} (3 x - 1)}{3} + \frac{11}{3} \cdot e^{3 x} + C}{e^{3 x}}$

Step 7.3.2.10

Combine $\frac{11}{3}$ and $e^{3 x}$ .

$y = \frac{\frac{2 e^{3 x} (3 x - 1)}{3} + \frac{11 e^{3 x}}{3} + C}{e^{3 x}}$

Step 7.3.3

Combine the numerators over the common denominator.

$y = \frac{C + \frac{2 e^{3 x} (3 x - 1) + 11 e^{3 x}}{3}}{e^{3 x}}$

Step 7.3.4

Simplify each term.

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Step 7.3.4.1

Apply the distributive property.

$y = \frac{C + \frac{2 e^{3 x} (3 x) + 2 e^{3 x} \cdot - 1 + 11 e^{3 x}}{3}}{e^{3 x}}$

Step 7.3.4.2

Rewrite using the commutative property of multiplication.

$y = \frac{C + \frac{2 \cdot 3 e^{3 x} x + 2 e^{3 x} \cdot - 1 + 11 e^{3 x}}{3}}{e^{3 x}}$

Step 7.3.4.3

Multiply $- 1$ by $2$ .

$y = \frac{C + \frac{2 \cdot 3 e^{3 x} x - 2 e^{3 x} + 11 e^{3 x}}{3}}{e^{3 x}}$

Step 7.3.4.4

Multiply $2$ by $3$ .

$y = \frac{C + \frac{6 e^{3 x} x - 2 e^{3 x} + 11 e^{3 x}}{3}}{e^{3 x}}$

Step 7.3.5

Simplify by adding terms.

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Step 7.3.5.1

Add $- 2 e^{3 x}$ and $11 e^{3 x}$ .

$y = \frac{C + \frac{6 e^{3 x} x + 9 e^{3 x}}{3}}{e^{3 x}}$

Step 7.3.5.2

Reorder factors in $6 e^{3 x} x + 9 e^{3 x}$ .

$y = \frac{C + \frac{6 x e^{3 x} + 9 e^{3 x}}{3}}{e^{3 x}}$

Step 7.3.6

Simplify each term.

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Step 7.3.6.1

Factor $3 e^{3 x}$ out of $6 x e^{3 x} + 9 e^{3 x}$ .

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Step 7.3.6.1.1

Factor $3 e^{3 x}$ out of $6 x e^{3 x}$ .

$y = \frac{C + \frac{3 e^{3 x} (2 x) + 9 e^{3 x}}{3}}{e^{3 x}}$

Step 7.3.6.1.2

Factor $3 e^{3 x}$ out of $9 e^{3 x}$ .

$y = \frac{C + \frac{3 e^{3 x} (2 x) + 3 e^{3 x} (3)}{3}}{e^{3 x}}$

Step 7.3.6.1.3

Factor $3 e^{3 x}$ out of $3 e^{3 x} (2 x) + 3 e^{3 x} (3)$ .

$y = \frac{C + \frac{3 e^{3 x} (2 x + 3)}{3}}{e^{3 x}}$

Step 7.3.6.2

Cancel the common factor of $3$ .

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Step 7.3.6.2.1

Cancel the common factor.

$y = \frac{C + \frac{3 e^{3 x} (2 x + 3)}{3}}{e^{3 x}}$

Step 7.3.6.2.2

Divide $e^{3 x} (2 x + 3)$ by $1$ .

$y = \frac{C + e^{3 x} (2 x + 3)}{e^{3 x}}$

Step 7.3.6.3

Apply the distributive property.

$y = \frac{C + e^{3 x} (2 x) + e^{3 x} \cdot 3}{e^{3 x}}$

Step 7.3.6.4

Rewrite using the commutative property of multiplication.

$y = \frac{C + 2 e^{3 x} x + e^{3 x} \cdot 3}{e^{3 x}}$

Step 7.3.6.5

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

$y = \frac{C + 2 e^{3 x} x + 3 e^{3 x}}{e^{3 x}}$

Step 7.3.7

Reorder factors in $C + 2 e^{3 x} x + 3 e^{3 x}$ .

$y = \frac{C + 2 x e^{3 x} + 3 e^{3 x}}{e^{3 x}}$