Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Regroup factors.

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

Step 1.2

Multiply both sides by $y$ .

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

Step 1.3

Simplify.

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Step 1.3.1

Simplify the denominator.

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Step 1.3.1.1

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

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

Step 1.3.1.2

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 $a = 1$ and $b = x$ .

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

Step 1.3.1.3

Simplify.

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Step 1.3.1.3.1

One to any power is one.

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

Step 1.3.1.3.2

Multiply $x$ by $1$ .

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

Step 1.3.2

Combine.

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

Step 1.3.3

Cancel the common factor of $y$ .

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Step 1.3.3.1

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

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

Step 1.3.3.2

Cancel the common factor.

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

Step 1.3.3.3

Rewrite the expression.

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

Step 1.3.4

Multiply $3$ by $1$ .

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

Step 1.4

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

By the Power Rule, the integral of $y$ with respect to $y$ is $\frac{1}{2} y^{2}$ .

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

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

Step 2.3.2

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

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Step 2.3.2.1

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

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Step 2.3.2.1.1

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

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

Step 2.3.2.1.2

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

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

Step 2.3.2.1.3

Differentiate.

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Step 2.3.2.1.3.1

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

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

Step 2.3.2.1.3.2

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

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

Step 2.3.2.1.3.3

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

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

Step 2.3.2.1.3.4

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

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

Step 2.3.2.1.3.5

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

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

Step 2.3.2.1.3.6

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

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

Step 2.3.2.1.3.7

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

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

Step 2.3.2.1.3.8

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

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

Step 2.3.2.1.3.9

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

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

Step 2.3.2.1.3.10

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

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

Step 2.3.2.1.3.11

Simplify the expression.

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Step 2.3.2.1.3.11.1

Multiply $- 1$ by $1$ .

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

Step 2.3.2.1.3.11.2

Move $- 1$ to the left of $1 + x + x^{2}$ .

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

Step 2.3.2.1.3.11.3

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

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

Step 2.3.2.1.4

Simplify.

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Step 2.3.2.1.4.1

Apply the distributive property.

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

Step 2.3.2.1.4.2

Apply the distributive property.

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

Step 2.3.2.1.4.3

Apply the distributive property.

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

Step 2.3.2.1.4.4

Apply the distributive property.

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

Step 2.3.2.1.4.5

Combine terms.

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Step 2.3.2.1.4.5.1

Multiply $1$ by $1$ .

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

Step 2.3.2.1.4.5.2

Multiply $- 1$ by $1$ .

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

Step 2.3.2.1.4.5.3

Multiply $2$ by $1$ .

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

Step 2.3.2.1.4.5.4

Multiply $2$ by $- 1$ .

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

Step 2.3.2.1.4.5.5

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

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

Step 2.3.2.1.4.5.6

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

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

Step 2.3.2.1.4.5.7

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

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

Step 2.3.2.1.4.5.8

Add $1$ and $1$ .

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

Step 2.3.2.1.4.5.9

Add $- x$ and $2 x$ .

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

Step 2.3.2.1.4.5.10

Multiply $- 1$ by $1$ .

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

Step 2.3.2.1.4.5.11

Subtract $1$ from $1$ .

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

Step 2.3.2.1.4.5.12

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

$x - 2 x^{2} - x - x^{2}$

Step 2.3.2.1.4.5.13

Subtract $x$ from $x$ .

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

Step 2.3.2.1.4.5.14

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

$- 2 x^{2} - x^{2}$

Step 2.3.2.1.4.5.15

Subtract $x^{2}$ from $- 2 x^{2}$ .

$- 3 x^{2}$

Step 2.3.2.2

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

$\frac{1}{2} y^{2} + C_{1} = 3 \int \frac{1}{u} \cdot \frac{1}{- 3} d u$

Step 2.3.3

Simplify.

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Step 2.3.3.1

Move the negative in front of the fraction.

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

Step 2.3.3.2

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

$\frac{1}{2} y^{2} + C_{1} = 3 \int - \frac{1}{u \cdot 3} d u$

Step 2.3.3.3

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

$\frac{1}{2} y^{2} + C_{1} = 3 \int - \frac{1}{3 u} d u$

Step 2.3.4

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

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

Step 2.3.5

Multiply $- 1$ by $3$ .

$\frac{1}{2} y^{2} + C_{1} = - 3 \int \frac{1}{3 u} d u$

Step 2.3.6

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

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

Step 2.3.7

Simplify.

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Step 2.3.7.1

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

$\frac{1}{2} y^{2} + C_{1} = \frac{- 3}{3} \int \frac{1}{u} d u$

Step 2.3.7.2

Cancel the common factor of $- 3$ and $3$ .

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Step 2.3.7.2.1

Factor $3$ out of $- 3$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{3 \cdot - 1}{3} \int \frac{1}{u} d u$

Step 2.3.7.2.2

Cancel the common factors.

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Step 2.3.7.2.2.1

Factor $3$ out of $3$ .

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

Step 2.3.7.2.2.2

Cancel the common factor.

$\frac{1}{2} y^{2} + C_{1} = \frac{3 \cdot - 1}{3 \cdot 1} \int \frac{1}{u} d u$

Step 2.3.7.2.2.3

Rewrite the expression.

$\frac{1}{2} y^{2} + C_{1} = \frac{- 1}{1} \int \frac{1}{u} d u$

Step 2.3.7.2.2.4

Divide $- 1$ by $1$ .

$\frac{1}{2} y^{2} + C_{1} = - \int \frac{1}{u} d u$

Step 2.3.8

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

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

Step 2.3.9

Simplify.

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

Step 2.3.10

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

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

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

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $2 (\frac{1}{2} y^{2})$ .

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Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $2 (- \ln (| (1 - x) (1 + x + x^{2}) |) + C)$ .

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Step 3.2.2.1.1

Simplify each term.

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Step 3.2.2.1.1.1

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

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

Step 3.2.2.1.1.2

Simplify each term.

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Step 3.2.2.1.1.2.1

Multiply $1$ by $1$ .

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

Step 3.2.2.1.1.2.2

Multiply $x$ by $1$ .

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

Step 3.2.2.1.1.2.3

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

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

Step 3.2.2.1.1.2.4

Multiply $- 1$ by $1$ .

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

Step 3.2.2.1.1.2.5

Multiply $x$ by $x$ by adding the exponents.

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Step 3.2.2.1.1.2.5.1

Move $x$ .

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

Step 3.2.2.1.1.2.5.2

Multiply $x$ by $x$ .

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

Step 3.2.2.1.1.2.6

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

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Step 3.2.2.1.1.2.6.1

Move $x^{2}$ .

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

Step 3.2.2.1.1.2.6.2

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

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Step 3.2.2.1.1.2.6.2.1

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

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

Step 3.2.2.1.1.2.6.2.2

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

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

Step 3.2.2.1.1.2.6.3

Add $2$ and $1$ .

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

Step 3.2.2.1.1.3

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

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Step 3.2.2.1.1.3.1

Subtract $x$ from $x$ .

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

Step 3.2.2.1.1.3.2

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

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

Step 3.2.2.1.1.3.3

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

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

Step 3.2.2.1.1.3.4

Add $1$ and $0$ .

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

Step 3.2.2.1.2

Simplify by multiplying through.

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Step 3.2.2.1.2.1

Apply the distributive property.

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

Step 3.2.2.1.2.2

Multiply $- 1$ by $2$ .

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

Step 3.3

Simplify $- 2 \ln (| 1 - x^{3} |)$ by moving $2$ inside the logarithm.

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

Step 3.4

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

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

Step 3.5

Remove the absolute value in ${| 1 - x^{3} |}^{2}$ because exponentiations with even powers are always positive.

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

Step 3.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.6.1

First, use the positive value of the $\pm$ to find the first solution.

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

Step 3.6.2

Next, use the negative value of the $\pm$ to find the second solution.

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

Step 3.6.3

The complete solution is the result of both the positive and negative portions of the solution.

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

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

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

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

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

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

Step 4

Simplify the constant of integration.

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

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