Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Regroup factors.

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

Step 1.2

Multiply both sides by $y$ .

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

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 $1^{3}$ .

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

Step 1.3.1.2

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 $b = x$ .

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

Step 1.3.1.3

Simplify.

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Step 1.3.1.3.1

One to any power is one.

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

Step 1.3.1.3.2

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

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

Step 1.3.2

Combine.

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

Step 1.3.3

Cancel the common factor of $y$ .

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Step 1.3.3.1

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

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

Step 1.3.3.2

Cancel the common factor.

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

Step 1.3.3.3

Rewrite the expression.

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

Step 1.3.4

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

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

Step 1.4

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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 2.3.1.1

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

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Step 2.3.1.1.1

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

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

Step 2.3.1.1.2

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 2.3.1.1.3

Differentiate.

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Step 2.3.1.1.3.1

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 2.3.1.1.3.2

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 2.3.1.1.3.3

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 2.3.1.1.3.4

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

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

Step 2.3.1.1.3.5

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

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

Step 2.3.1.1.3.6

Multiply $- 1$ by $1$ .

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

Step 2.3.1.1.3.7

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 2.3.1.1.3.8

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 2.3.1.1.3.9

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 2.3.1.1.3.10

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

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

Step 2.3.1.1.3.11

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

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

Step 2.3.1.1.3.12

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

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

Step 2.3.1.1.4

Simplify.

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Step 2.3.1.1.4.1

Apply the distributive property.

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

Step 2.3.1.1.4.2

Apply the distributive property.

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

Step 2.3.1.1.4.3

Apply the distributive property.

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

Step 2.3.1.1.4.4

Combine terms.

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Step 2.3.1.1.4.4.1

Multiply $- 1$ by $1$ .

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

Step 2.3.1.1.4.4.2

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

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

Step 2.3.1.1.4.4.3

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

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

Step 2.3.1.1.4.4.4

Multiply $2$ by $1$ .

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

Step 2.3.1.1.4.4.5

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

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

Step 2.3.1.1.4.4.6

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

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

Step 2.3.1.1.4.4.7

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

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

Step 2.3.1.1.4.4.8

Add $1$ and $1$ .

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

Step 2.3.1.1.4.4.9

Add $- x$ and $2 x$ .

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

Step 2.3.1.1.4.4.10

Add $- 1$ and $1$ .

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

Step 2.3.1.1.4.4.11

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

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

Step 2.3.1.1.4.4.12

Subtract $x$ from $x$ .

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

Step 2.3.1.1.4.4.13

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

$2 x^{2} + x^{2}$

Step 2.3.1.1.4.4.14

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

$3 x^{2}$

Step 2.3.1.2

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

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

Step 2.3.2

Simplify.

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Step 2.3.2.1

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

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

Step 2.3.2.2

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

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Simplify.

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

Step 2.3.6

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

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

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

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 $y^{2}$ .

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

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.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

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

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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.

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

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 $1$ .

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

Step 3.2.2.1.1.2.2

Multiply $- x$ by $1$ .

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

Step 3.2.2.1.1.2.3

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

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

Step 3.2.2.1.1.2.4

Multiply $x$ by $1$ .

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

Step 3.2.2.1.1.2.5

Rewrite using the commutative property of multiplication.

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

Step 3.2.2.1.1.2.6

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

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Step 3.2.2.1.1.2.6.1

Move $x$ .

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

Step 3.2.2.1.1.2.6.2

Multiply $x$ by $x$ .

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

Step 3.2.2.1.1.2.7

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

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Step 3.2.2.1.1.2.7.1

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

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Step 3.2.2.1.1.2.7.1.1

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

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

Step 3.2.2.1.1.2.7.1.2

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

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

Step 3.2.2.1.1.2.7.2

Add $1$ and $2$ .

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

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

Add $- x$ and $x$ .

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

Step 3.2.2.1.1.3.2

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

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

Step 3.2.2.1.1.3.3

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

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

Step 3.2.2.1.1.3.4

Add $1$ and $0$ .

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

Step 3.2.2.1.1.4

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

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

Step 3.2.2.1.2

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

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

Step 3.2.2.1.3

Simplify terms.

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Step 3.2.2.1.3.1

Combine $C$ and $\frac{3}{3}$ .

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

Step 3.2.2.1.3.2

Combine the numerators over the common denominator.

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

Step 3.2.2.1.4

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

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

Step 3.2.2.1.5

Combine $2$ and $\frac{\ln (| 1 + x^{3} |) + 3 C}{3}$ .

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

Step 3.3

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

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

Step 3.4

Simplify $\pm \sqrt{\frac{2 (\ln (| 1 + x^{3} |) + 3 C)}{3}}$ .

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Step 3.4.1

Rewrite $\sqrt{\frac{2 (\ln (| 1 + x^{3} |) + 3 C)}{3}}$ as $\frac{\sqrt{2 (\ln (| 1 + x^{3} |) + 3 C)}}{\sqrt{3}}$ .

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

Step 3.4.2

Multiply $\frac{\sqrt{2 (\ln (| 1 + x^{3} |) + 3 C)}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 3.4.3

Combine and simplify the denominator.

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Step 3.4.3.1

Multiply $\frac{\sqrt{2 (\ln (| 1 + x^{3} |) + 3 C)}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 3.4.3.2

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 3.4.3.3

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 3.4.3.4

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

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

Step 3.4.3.5

Add $1$ and $1$ .

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

Step 3.4.3.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.4.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 3.4.3.6.2

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

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

Step 3.4.3.6.3

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

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

Step 3.4.3.6.4

Cancel the common factor of $2$ .

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Step 3.4.3.6.4.1

Cancel the common factor.

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

Step 3.4.3.6.4.2

Rewrite the expression.

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

Step 3.4.3.6.5

Evaluate the exponent.

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

Step 3.4.4

Simplify the numerator.

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Step 3.4.4.1

Combine using the product rule for radicals.

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

Step 3.4.4.2

Multiply $3$ by $2$ .

$y = \pm \frac{\sqrt{6 (\ln (| 1 + x^{3} |) + 3 C)}}{3}$

Step 3.5

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

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Step 3.5.1

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

$y = \frac{\sqrt{6 (\ln (| 1 + x^{3} |) + 3 C)}}{3}$

Step 3.5.2

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

$y = - \frac{\sqrt{6 (\ln (| 1 + x^{3} |) + 3 C)}}{3}$

Step 3.5.3

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

$y = \frac{\sqrt{6 (\ln (| 1 + x^{3} |) + 3 C)}}{3}$

$y = - \frac{\sqrt{6 (\ln (| 1 + x^{3} |) + 3 C)}}{3}$

$y = \frac{\sqrt{6 (\ln (| 1 + x^{3} |) + 3 C)}}{3}$

$y = - \frac{\sqrt{6 (\ln (| 1 + x^{3} |) + 3 C)}}{3}$

$y = \frac{\sqrt{6 (\ln (| 1 + x^{3} |) + 3 C)}}{3}$

$y = - \frac{\sqrt{6 (\ln (| 1 + x^{3} |) + 3 C)}}{3}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt{6 (\ln (| 1 + x^{3} |) + C)}}{3}$

$y = - \frac{\sqrt{6 (\ln (| 1 + x^{3} |) + C)}}{3}$