Calculus Examples

$(1 + y^{3}) d x + x y^{2} d y = 0$

Step 1

Subtract $(1 + y^{3}) d x$ from both sides of the equation.

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

Step 2

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

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

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $x$ .

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Step 3.1.1

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

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

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

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

Step 3.3

Simplify the denominator.

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

Simplify.

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Step 3.3.3.1

One to any power is one.

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

Step 3.3.3.2

Rewrite $- 1 y$ as $- y$ .

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

Step 3.4

Rewrite using the commutative property of multiplication.

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

Step 3.5

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

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Step 3.5.1

Move the leading negative in $- \frac{1}{x (1 + y^{3})}$ into the numerator.

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

Step 3.5.2

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

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

Step 3.5.3

Cancel the common factor.

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

Step 3.5.4

Rewrite the expression.

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

Step 3.6

Move the negative in front of the fraction.

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

Integrate the left side.

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Step 4.2.1

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

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Step 4.2.1.1.1

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

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

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 $f (y) = 1 + y$ and $g (y) = 1 - y + y^{2}$ .

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

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 $1 - y + y^{2}$ with respect to $y$ is $\frac{d}{d y} [1] + \frac{d}{d y} [- y] + \frac{d}{d y} [y^{2}]$ .

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

Step 4.2.1.1.3.2

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

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

Step 4.2.1.1.3.3

Add $0$ and $\frac{d}{d y} [- y]$ .

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

Step 4.2.1.1.3.4

Since $- 1$ is constant with respect to $y$ , the derivative of $- y$ with respect to $y$ is $- \frac{d}{d y} [y]$ .

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

Step 4.2.1.1.3.5

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

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

Step 4.2.1.1.3.6

Multiply $- 1$ by $1$ .

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

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 $n = 2$ .

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

Step 4.2.1.1.3.8

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

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

Step 4.2.1.1.3.9

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

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

Step 4.2.1.1.3.10

Add $0$ and $\frac{d}{d y} [y]$ .

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

Step 4.2.1.1.3.11

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

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

Step 4.2.1.1.3.12

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

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

Step 4.2.1.1.4

Simplify.

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Step 4.2.1.1.4.1

Apply the distributive property.

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

Step 4.2.1.1.4.2

Apply the distributive property.

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

Step 4.2.1.1.4.3

Apply the distributive property.

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

Step 4.2.1.1.4.4

Combine terms.

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Step 4.2.1.1.4.4.1

Multiply $- 1$ by $1$ .

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

Step 4.2.1.1.4.4.2

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

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

Step 4.2.1.1.4.4.3

Rewrite $- 1 y$ as $- y$ .

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

Step 4.2.1.1.4.4.4

Multiply $2$ by $1$ .

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

Step 4.2.1.1.4.4.5

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

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

Step 4.2.1.1.4.4.6

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

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

Step 4.2.1.1.4.4.7

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

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

Step 4.2.1.1.4.4.8

Add $1$ and $1$ .

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

Step 4.2.1.1.4.4.9

Add $- y$ and $2 y$ .

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

Step 4.2.1.1.4.4.10

Add $- 1$ and $1$ .

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

Step 4.2.1.1.4.4.11

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

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

Step 4.2.1.1.4.4.12

Subtract $y$ from $y$ .

$2 y^{2} + 0 + y^{2}$

Step 4.2.1.1.4.4.13

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

$2 y^{2} + y^{2}$

Step 4.2.1.1.4.4.14

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

$3 y^{2}$

Step 4.2.1.2

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

$\int \frac{1}{u} \cdot \frac{1}{3} d u = \int - \frac{1}{x} d x$

Step 4.2.2

Simplify.

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Step 4.2.2.1

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

$\int \frac{1}{u \cdot 3} d u = \int - \frac{1}{x} d x$

Step 4.2.2.2

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

$\int \frac{1}{3 u} d u = \int - \frac{1}{x} d x$

Step 4.2.3

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

$\frac{1}{3} \int \frac{1}{u} d u = \int - \frac{1}{x} d x$

Step 4.2.4

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

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

Step 4.2.5

Simplify.

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

Step 4.2.6

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

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

Step 4.3

Integrate the right side.

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

Simplify.

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

Step 4.4

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

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

Step 5

Solve for $y$ .

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Step 5.1

Multiply both sides of the equation by $3$ .

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

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

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Step 5.2.1.1.1

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

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

Step 5.2.1.1.2

Simplify terms.

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Step 5.2.1.1.2.1

Simplify each term.

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Step 5.2.1.1.2.1.1

Multiply $1$ by $1$ .

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

Step 5.2.1.1.2.1.2

Multiply $- y$ by $1$ .

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

Step 5.2.1.1.2.1.3

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

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

Step 5.2.1.1.2.1.4

Multiply $y$ by $1$ .

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

Step 5.2.1.1.2.1.5

Rewrite using the commutative property of multiplication.

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

Step 5.2.1.1.2.1.6

Multiply $y$ by $y$ by adding the exponents.

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Step 5.2.1.1.2.1.6.1

Move $y$ .

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

Step 5.2.1.1.2.1.6.2

Multiply $y$ by $y$ .

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

Step 5.2.1.1.2.1.7

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

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Step 5.2.1.1.2.1.7.1

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

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Step 5.2.1.1.2.1.7.1.1

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

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

Step 5.2.1.1.2.1.7.1.2

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

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

Step 5.2.1.1.2.1.7.2

Add $1$ and $2$ .

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

Step 5.2.1.1.2.2

Simplify terms.

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Step 5.2.1.1.2.2.1

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

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Step 5.2.1.1.2.2.1.1

Add $- y$ and $y$ .

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

Step 5.2.1.1.2.2.1.2

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

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

Step 5.2.1.1.2.2.1.3

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

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

Step 5.2.1.1.2.2.1.4

Add $1$ and $0$ .

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

Step 5.2.1.1.2.2.2

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

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

Step 5.2.1.1.2.2.3

Cancel the common factor of $3$ .

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Step 5.2.1.1.2.2.3.1

Cancel the common factor.

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

Step 5.2.1.1.2.2.3.2

Rewrite the expression.

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

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $3 (- \ln (| x |) + C)$ .

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Step 5.2.2.1.1

Apply the distributive property.

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

Step 5.2.2.1.2

Multiply $- 1$ by $3$ .

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

Step 5.3

Move all the terms containing a logarithm to the left side of the equation.

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

Step 5.4

Simplify the left side.

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Step 5.4.1

Simplify $\ln (| 1 + y^{3} |) + 3 \ln (| x |)$ .

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Step 5.4.1.1

Simplify $3 \ln (| x |)$ by moving $3$ inside the logarithm.

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

Step 5.4.1.2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

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

Step 5.5

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

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

Step 5.6

Rewrite $\ln (| 1 + y^{3} | {| x |}^{3}) = 3 C$ 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 $b^{y} = x$ .

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

Step 5.7

Solve for $y$ .

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Step 5.7.1

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

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

Step 5.7.2

Divide each term in $| 1 + y^{3} | {| x |}^{3} = e^{3 C}$ by ${| x |}^{3}$ and simplify.

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Step 5.7.2.1

Divide each term in $| 1 + y^{3} | {| x |}^{3} = e^{3 C}$ by ${| x |}^{3}$ .

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

Step 5.7.2.2

Simplify the left side.

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Step 5.7.2.2.1

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

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Step 5.7.2.2.1.1

Cancel the common factor.

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

Step 5.7.2.2.1.2

Divide $| 1 + y^{3} |$ by $1$ .

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

Step 5.7.3

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

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

Step 5.7.4

Subtract $1$ from both sides of the equation.

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

Step 5.7.5

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

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

Step 6

Group the constant terms together.

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Step 6.1

Simplify the constant of integration.

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

Step 6.2

Combine constants with the plus or minus.

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