Calculus Examples

$(x^{2} + 1) (y^{3} - 1) d x = x^{2} y^{2} d y$

Step 1

Rewrite the equation.

$x^{2} y^{2} d y = (x^{2} + 1) (y^{3} - 1) d x$

Step 2

Multiply both sides by $\frac{1}{x^{2} (y^{3} - 1)}$ .

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

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $x^{2}$ .

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Step 3.1.1

Factor $x^{2}$ out of $x^{2} y^{2}$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Combine $\frac{1}{y^{3} - 1}$ and $y^{2}$ .

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

Step 3.3

Simplify the denominator.

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Step 3.3.1

Rewrite $1$ as $1^{3}$ .

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

Step 3.3.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = y$ and $b = 1$ .

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

Step 3.3.3

Simplify.

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Step 3.3.3.1

Multiply $y$ by $1$ .

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

Step 3.3.3.2

One to any power is one.

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

Step 3.4

Cancel the common factor of $y^{3} - 1$ .

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Step 3.4.1

Factor $y^{3} - 1$ out of $x^{2} (y^{3} - 1)$ .

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

Step 3.4.2

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

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

Step 3.4.3

Cancel the common factor.

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

Step 3.4.4

Rewrite the expression.

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

Step 3.5

Multiply $\frac{1}{x^{2}}$ by $x^{2} + 1$ .

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

Integrate the left side.

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Step 4.2.1

Let $u = (y - 1) (y^{2} + y + 1)$ . Then $d u = 3 y^{2} d y$ , so $\frac{1}{3} d u = y^{2} d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.2.1.1

Let $u = (y - 1) (y^{2} + y + 1)$ . Find $\frac{d u}{d y}$ .

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Step 4.2.1.1.1

Differentiate $(y - 1) (y^{2} + y + 1)$ .

$\frac{d}{d y} [(y - 1) (y^{2} + y + 1)]$

Step 4.2.1.1.2

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = y - 1$ and $g (y) = y^{2} + y + 1$ .

$(y - 1) \frac{d}{d y} [y^{2} + y + 1] + (y^{2} + y + 1) \frac{d}{d y} [y - 1]$

Step 4.2.1.1.3

Differentiate.

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Step 4.2.1.1.3.1

By the Sum Rule, the derivative of $y^{2} + y + 1$ with respect to $y$ is $\frac{d}{d y} [y^{2}] + \frac{d}{d y} [y] + \frac{d}{d y} [1]$ .

$(y - 1) (\frac{d}{d y} [y^{2}] + \frac{d}{d y} [y] + \frac{d}{d y} [1]) + (y^{2} + y + 1) \frac{d}{d y} [y - 1]$

Step 4.2.1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 2$ .

$(y - 1) (2 y + \frac{d}{d y} [y] + \frac{d}{d y} [1]) + (y^{2} + y + 1) \frac{d}{d y} [y - 1]$

Step 4.2.1.1.3.3

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

$(y - 1) (2 y + 1 + \frac{d}{d y} [1]) + (y^{2} + y + 1) \frac{d}{d y} [y - 1]$

Step 4.2.1.1.3.4

Since $1$ is constant with respect to $y$ , the derivative of $1$ with respect to $y$ is $0$ .

$(y - 1) (2 y + 1 + 0) + (y^{2} + y + 1) \frac{d}{d y} [y - 1]$

Step 4.2.1.1.3.5

Add $2 y + 1$ and $0$ .

$(y - 1) (2 y + 1) + (y^{2} + y + 1) \frac{d}{d y} [y - 1]$

Step 4.2.1.1.3.6

By the Sum Rule, the derivative of $y - 1$ with respect to $y$ is $\frac{d}{d y} [y] + \frac{d}{d y} [- 1]$ .

$(y - 1) (2 y + 1) + (y^{2} + y + 1) (\frac{d}{d y} [y] + \frac{d}{d y} [- 1])$

Step 4.2.1.1.3.7

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

$(y - 1) (2 y + 1) + (y^{2} + y + 1) (1 + \frac{d}{d y} [- 1])$

Step 4.2.1.1.3.8

Since $- 1$ is constant with respect to $y$ , the derivative of $- 1$ with respect to $y$ is $0$ .

$(y - 1) (2 y + 1) + (y^{2} + y + 1) (1 + 0)$

Step 4.2.1.1.3.9

Simplify the expression.

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Step 4.2.1.1.3.9.1

Add $1$ and $0$ .

$(y - 1) (2 y + 1) + (y^{2} + y + 1) \cdot 1$

Step 4.2.1.1.3.9.2

Multiply $y^{2} + y + 1$ by $1$ .

$(y - 1) (2 y + 1) + y^{2} + y + 1$

Step 4.2.1.1.4

Simplify.

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Step 4.2.1.1.4.1

Apply the distributive property.

$(y - 1) (2 y) + (y - 1) \cdot 1 + y^{2} + y + 1$

Step 4.2.1.1.4.2

Apply the distributive property.

$y (2 y) - 1 (2 y) + (y - 1) \cdot 1 + y^{2} + y + 1$

Step 4.2.1.1.4.3

Apply the distributive property.

$y (2 y) - 1 (2 y) + y \cdot 1 - 1 \cdot 1 + y^{2} + y + 1$

Step 4.2.1.1.4.4

Combine terms.

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Step 4.2.1.1.4.4.1

Raise $y$ to the power of $1$ .

$2 (y^{1} y) - 1 (2 y) + y \cdot 1 - 1 \cdot 1 + y^{2} + y + 1$

Step 4.2.1.1.4.4.2

Raise $y$ to the power of $1$ .

$2 (y^{1} y^{1}) - 1 (2 y) + y \cdot 1 - 1 \cdot 1 + y^{2} + y + 1$

Step 4.2.1.1.4.4.3

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

$2 y^{1 + 1} - 1 (2 y) + y \cdot 1 - 1 \cdot 1 + y^{2} + y + 1$

Step 4.2.1.1.4.4.4

Add $1$ and $1$ .

$2 y^{2} - 1 (2 y) + y \cdot 1 - 1 \cdot 1 + y^{2} + y + 1$

Step 4.2.1.1.4.4.5

Multiply $2$ by $- 1$ .

$2 y^{2} - 2 y + y \cdot 1 - 1 \cdot 1 + y^{2} + y + 1$

Step 4.2.1.1.4.4.6

Multiply $y$ by $1$ .

$2 y^{2} - 2 y + y - 1 \cdot 1 + y^{2} + y + 1$

Step 4.2.1.1.4.4.7

Multiply $- 1$ by $1$ .

$2 y^{2} - 2 y + y - 1 + y^{2} + y + 1$

Step 4.2.1.1.4.4.8

Add $- 2 y$ and $y$ .

$2 y^{2} - y - 1 + y^{2} + y + 1$

Step 4.2.1.1.4.4.9

Add $2 y^{2}$ and $y^{2}$ .

$3 y^{2} - y - 1 + y + 1$

Step 4.2.1.1.4.4.10

Add $- y$ and $y$ .

$3 y^{2} + 0 - 1 + 1$

Step 4.2.1.1.4.4.11

Add $3 y^{2}$ and $0$ .

$3 y^{2} - 1 + 1$

Step 4.2.1.1.4.4.12

Add $- 1$ and $1$ .

$3 y^{2} + 0$

Step 4.2.1.1.4.4.13

Add $3 y^{2}$ and $0$ .

$3 y^{2}$

Step 4.2.1.2

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

$\int \frac{1}{u} \cdot \frac{1}{3} d u = \int \frac{x^{2} + 1}{x^{2}} d x$

Step 4.2.2

Simplify.

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Step 4.2.2.1

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

$\int \frac{1}{u \cdot 3} d u = \int \frac{x^{2} + 1}{x^{2}} d x$

Step 4.2.2.2

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

$\int \frac{1}{3 u} d u = \int \frac{x^{2} + 1}{x^{2}} d x$

Step 4.2.3

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

$\frac{1}{3} \int \frac{1}{u} d u = \int \frac{x^{2} + 1}{x^{2}} d x$

Step 4.2.4

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$\frac{1}{3} (\ln (| u |) + C_{1}) = \int \frac{x^{2} + 1}{x^{2}} d x$

Step 4.2.5

Simplify.

$\frac{1}{3} \ln (| u |) + C_{1} = \int \frac{x^{2} + 1}{x^{2}} d x$

Step 4.2.6

Replace all occurrences of $u$ with $(y - 1) (y^{2} + y + 1)$ .

$\frac{1}{3} \ln (| (y - 1) (y^{2} + y + 1) |) + C_{1} = \int \frac{x^{2} + 1}{x^{2}} d x$

Step 4.3

Integrate the right side.

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Step 4.3.1

Apply basic rules of exponents.

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Step 4.3.1.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{3} \ln (| (y - 1) (y^{2} + y + 1) |) + C_{1} = \int (x^{2} + 1) {(x^{2})}^{- 1} d x$

Step 4.3.1.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 4.3.1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{3} \ln (| (y - 1) (y^{2} + y + 1) |) + C_{1} = \int (x^{2} + 1) x^{2 \cdot - 1} d x$

Step 4.3.1.2.2

Multiply $2$ by $- 1$ .

$\frac{1}{3} \ln (| (y - 1) (y^{2} + y + 1) |) + C_{1} = \int (x^{2} + 1) x^{- 2} d x$

Step 4.3.2

Multiply $(x^{2} + 1) x^{- 2}$ .

$\frac{1}{3} \ln (| (y - 1) (y^{2} + y + 1) |) + C_{1} = \int x^{2} x^{- 2} + 1 x^{- 2} d x$

Step 4.3.3

Simplify.

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Step 4.3.3.1

Multiply $x^{2}$ by $x^{- 2}$ by adding the exponents.

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Step 4.3.3.1.1

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

$\frac{1}{3} \ln (| (y - 1) (y^{2} + y + 1) |) + C_{1} = \int x^{2 - 2} + 1 x^{- 2} d x$

Step 4.3.3.1.2

Subtract $2$ from $2$ .

$\frac{1}{3} \ln (| (y - 1) (y^{2} + y + 1) |) + C_{1} = \int x^{0} + 1 x^{- 2} d x$

Step 4.3.3.2

Simplify $x^{0}$ .

$\frac{1}{3} \ln (| (y - 1) (y^{2} + y + 1) |) + C_{1} = \int 1 + 1 x^{- 2} d x$

Step 4.3.3.3

Multiply $x^{- 2}$ by $1$ .

$\frac{1}{3} \ln (| (y - 1) (y^{2} + y + 1) |) + C_{1} = \int 1 + x^{- 2} d x$

Step 4.3.4

Split the single integral into multiple integrals.

$\frac{1}{3} \ln (| (y - 1) (y^{2} + y + 1) |) + C_{1} = \int d x + \int x^{- 2} d x$

Step 4.3.5

Apply the constant rule.

$\frac{1}{3} \ln (| (y - 1) (y^{2} + y + 1) |) + C_{1} = x + C_{2} + \int x^{- 2} d x$

Step 4.3.6

By the Power Rule, the integral of $x^{- 2}$ with respect to $x$ is $- x^{- 1}$ .

$\frac{1}{3} \ln (| (y - 1) (y^{2} + y + 1) |) + C_{1} = x + C_{2} - x^{- 1} + C_{3}$

Step 4.3.7

Simplify.

$\frac{1}{3} \ln (| (y - 1) (y^{2} + y + 1) |) + C_{1} = x - \frac{1}{x} + C_{4}$

Step 4.4

Group the constant of integration on the right side as $C$ .

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

Step 5

Solve for $y$ .

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Step 5.1

Multiply both sides of the equation by $3$ .

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

Step 5.2

Simplify both sides of the equation.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Simplify $3 (\frac{1}{3} \ln (| (y - 1) (y^{2} + y + 1) |))$ .

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Step 5.2.1.1.1

Expand $(y - 1) (y^{2} + y + 1)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 5.2.1.1.2

Simplify terms.

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Step 5.2.1.1.2.1

Combine the opposite terms in $y \cdot y^{2} + y \cdot y + y \cdot 1 - 1 y^{2} - 1 y - 1 \cdot 1$ .

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Step 5.2.1.1.2.1.1

Reorder the factors in the terms $y \cdot 1$ and $- 1 y$ .

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

Step 5.2.1.1.2.1.2

Subtract $1 y$ from $1 y$ .

$3 (\frac{1}{3} \ln (| y \cdot y^{2} + y \cdot y - 1 y^{2} + 0 - 1 \cdot 1 |)) = 3 (x - \frac{1}{x} + C)$

Step 5.2.1.1.2.1.3

Add $y \cdot y^{2} + y \cdot y - 1 y^{2}$ and $0$ .

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

Step 5.2.1.1.2.2

Simplify each term.

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Step 5.2.1.1.2.2.1

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 5.2.1.1.2.2.1.1

Multiply $y$ by $y^{2}$ .

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Step 5.2.1.1.2.2.1.1.1

Raise $y$ to the power of $1$ .

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

Step 5.2.1.1.2.2.1.1.2

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

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

Step 5.2.1.1.2.2.1.2

Add $1$ and $2$ .

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

Step 5.2.1.1.2.2.2

Multiply $y$ by $y$ .

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

Step 5.2.1.1.2.2.3

Rewrite $- 1 y^{2}$ as $- y^{2}$ .

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

Step 5.2.1.1.2.2.4

Multiply $- 1$ by $1$ .

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

Step 5.2.1.1.2.3

Simplify terms.

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Step 5.2.1.1.2.3.1

Combine the opposite terms in $y^{3} + y^{2} - y^{2} - 1$ .

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Step 5.2.1.1.2.3.1.1

Subtract $y^{2}$ from $y^{2}$ .

$3 (\frac{1}{3} \ln (| y^{3} + 0 - 1 |)) = 3 (x - \frac{1}{x} + C)$

Step 5.2.1.1.2.3.1.2

Add $y^{3}$ and $0$ .

$3 (\frac{1}{3} \ln (| y^{3} - 1 |)) = 3 (x - \frac{1}{x} + C)$

Step 5.2.1.1.2.3.2

Combine $\frac{1}{3}$ and $\ln (| y^{3} - 1 |)$ .

$3 \frac{\ln (| y^{3} - 1 |)}{3} = 3 (x - \frac{1}{x} + C)$

Step 5.2.1.1.2.3.3

Cancel the common factor of $3$ .

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Step 5.2.1.1.2.3.3.1

Cancel the common factor.

$3 \frac{\ln (| y^{3} - 1 |)}{3} = 3 (x - \frac{1}{x} + C)$

Step 5.2.1.1.2.3.3.2

Rewrite the expression.

$\ln (| y^{3} - 1 |) = 3 (x - \frac{1}{x} + C)$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $3 (x - \frac{1}{x} + C)$ .

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Step 5.2.2.1.1

Apply the distributive property.

$\ln (| y^{3} - 1 |) = 3 x + 3 (- \frac{1}{x}) + 3 C$

Step 5.2.2.1.2

Multiply $3 (- \frac{1}{x})$ .

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Step 5.2.2.1.2.1

Multiply $- 1$ by $3$ .

$\ln (| y^{3} - 1 |) = 3 x - 3 \frac{1}{x} + 3 C$

Step 5.2.2.1.2.2

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

$\ln (| y^{3} - 1 |) = 3 x + \frac{- 3}{x} + 3 C$

Step 5.2.2.1.3

Move the negative in front of the fraction.

$\ln (| y^{3} - 1 |) = 3 x - \frac{3}{x} + 3 C$

Step 5.3

To solve for $y$ , rewrite the equation using properties of logarithms.

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

Step 5.4

Rewrite $\ln (| y^{3} - 1 |) = 3 x - \frac{3}{x} + 3 C$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{3 x - \frac{3}{x} + 3 C} = | y^{3} - 1 |$

Step 5.5

Solve for $y$ .

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Step 5.5.1

Rewrite the equation as $| y^{3} - 1 | = e^{3 x - \frac{3}{x} + 3 C}$ .

$| y^{3} - 1 | = e^{3 x - \frac{3}{x} + 3 C}$

Step 5.5.2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$y^{3} - 1 = \pm e^{3 x - \frac{3}{x} + 3 C}$

Step 5.5.3

Add $1$ to both sides of the equation.

$y^{3} = \pm e^{3 x - \frac{3}{x} + 3 C} + 1$

Step 5.5.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \sqrt[3]{\pm e^{3 x - \frac{3}{x} + 3 C} + 1}$

Step 5.5.5

Simplify each term.

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Step 5.5.5.1

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

$y = \sqrt[3]{\pm e^{\frac{3 x \cdot x}{x} - \frac{3}{x} + 3 C} + 1}$

Step 5.5.5.2

Combine the numerators over the common denominator.

$y = \sqrt[3]{\pm e^{\frac{3 x \cdot x - 3}{x} + 3 C} + 1}$

Step 5.5.5.3

Simplify the numerator.

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Step 5.5.5.3.1

Factor $3$ out of $3 x \cdot x - 3$ .

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Step 5.5.5.3.1.1

Factor $3$ out of $3 x \cdot x$ .

$y = \sqrt[3]{\pm e^{\frac{3 (x \cdot x) - 3}{x} + 3 C} + 1}$

Step 5.5.5.3.1.2

Factor $3$ out of $- 3$ .

$y = \sqrt[3]{\pm e^{\frac{3 (x \cdot x) + 3 (- 1)}{x} + 3 C} + 1}$

Step 5.5.5.3.1.3

Factor $3$ out of $3 (x \cdot x) + 3 (- 1)$ .

$y = \sqrt[3]{\pm e^{\frac{3 (x \cdot x - 1)}{x} + 3 C} + 1}$

Step 5.5.5.3.2

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

$y = \sqrt[3]{\pm e^{\frac{3 (x^{1} x - 1)}{x} + 3 C} + 1}$

Step 5.5.5.3.3

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

$y = \sqrt[3]{\pm e^{\frac{3 (x^{1} x^{1} - 1)}{x} + 3 C} + 1}$

Step 5.5.5.3.4

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

$y = \sqrt[3]{\pm e^{\frac{3 (x^{1 + 1} - 1)}{x} + 3 C} + 1}$

Step 5.5.5.3.5

Add $1$ and $1$ .

$y = \sqrt[3]{\pm e^{\frac{3 (x^{2} - 1)}{x} + 3 C} + 1}$

Step 5.5.5.3.6

Rewrite $1$ as $1^{2}$ .

$y = \sqrt[3]{\pm e^{\frac{3 (x^{2} - 1^{2})}{x} + 3 C} + 1}$

Step 5.5.5.3.7

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$y = \sqrt[3]{\pm e^{\frac{3 (x + 1) (x - 1)}{x} + 3 C} + 1}$

Step 5.5.5.4

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

$y = \sqrt[3]{\pm e^{\frac{3 (x + 1) (x - 1)}{x} + \frac{3 C x}{x}} + 1}$

Step 5.5.5.5

Combine the numerators over the common denominator.

$y = \sqrt[3]{\pm e^{\frac{3 (x + 1) (x - 1) + 3 C x}{x}} + 1}$

Step 5.5.5.6

Simplify the numerator.

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Step 5.5.5.6.1

Factor $3$ out of $3 (x + 1) (x - 1) + 3 C x$ .

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Step 5.5.5.6.1.1

Factor $3$ out of $3 (x + 1) (x - 1)$ .

$y = \sqrt[3]{\pm e^{\frac{3 ((x + 1) (x - 1)) + 3 C x}{x}} + 1}$

Step 5.5.5.6.1.2

Factor $3$ out of $3 C x$ .

$y = \sqrt[3]{\pm e^{\frac{3 ((x + 1) (x - 1)) + 3 (C x)}{x}} + 1}$

Step 5.5.5.6.1.3

Factor $3$ out of $3 ((x + 1) (x - 1)) + 3 (C x)$ .

$y = \sqrt[3]{\pm e^{\frac{3 ((x + 1) (x - 1) + C x)}{x}} + 1}$

Step 5.5.5.6.2

Expand $(x + 1) (x - 1)$ using the FOIL Method.

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Step 5.5.5.6.2.1

Apply the distributive property.

$y = \sqrt[3]{\pm e^{\frac{3 (x (x - 1) + 1 (x - 1) + C x)}{x}} + 1}$

Step 5.5.5.6.2.2

Apply the distributive property.

$y = \sqrt[3]{\pm e^{\frac{3 (x \cdot x + x \cdot - 1 + 1 (x - 1) + C x)}{x}} + 1}$

Step 5.5.5.6.2.3

Apply the distributive property.

$y = \sqrt[3]{\pm e^{\frac{3 (x \cdot x + x \cdot - 1 + 1 x + 1 \cdot - 1 + C x)}{x}} + 1}$

Step 5.5.5.6.3

Simplify and combine like terms.

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Step 5.5.5.6.3.1

Simplify each term.

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Step 5.5.5.6.3.1.1

Multiply $x$ by $x$ .

$y = \sqrt[3]{\pm e^{\frac{3 (x^{2} + x \cdot - 1 + 1 x + 1 \cdot - 1 + C x)}{x}} + 1}$

Step 5.5.5.6.3.1.2

Move $- 1$ to the left of $x$ .

$y = \sqrt[3]{\pm e^{\frac{3 (x^{2} - 1 \cdot x + 1 x + 1 \cdot - 1 + C x)}{x}} + 1}$

Step 5.5.5.6.3.1.3

Rewrite $- 1 x$ as $- x$ .

$y = \sqrt[3]{\pm e^{\frac{3 (x^{2} - x + 1 x + 1 \cdot - 1 + C x)}{x}} + 1}$

Step 5.5.5.6.3.1.4

Multiply $x$ by $1$ .

$y = \sqrt[3]{\pm e^{\frac{3 (x^{2} - x + x + 1 \cdot - 1 + C x)}{x}} + 1}$

Step 5.5.5.6.3.1.5

Multiply $- 1$ by $1$ .

$y = \sqrt[3]{\pm e^{\frac{3 (x^{2} - x + x - 1 + C x)}{x}} + 1}$

Step 5.5.5.6.3.2

Add $- x$ and $x$ .

$y = \sqrt[3]{\pm e^{\frac{3 (x^{2} + 0 - 1 + C x)}{x}} + 1}$

Step 5.5.5.6.3.3

Add $x^{2}$ and $0$ .

$y = \sqrt[3]{\pm e^{\frac{3 (x^{2} - 1 + C x)}{x}} + 1}$