Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Divide each term in $\sqrt{1 + x^{3}} \frac{d y}{d x} = x^{2} y + x^{2}$ by $\sqrt{1 + x^{3}}$ and simplify.

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Step 1.1.1

Divide each term in $\sqrt{1 + x^{3}} \frac{d y}{d x} = x^{2} y + x^{2}$ by $\sqrt{1 + x^{3}}$ .

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

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of $\sqrt{1 + x^{3}}$ .

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Step 1.1.2.1.1

Cancel the common factor.

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

Step 1.1.2.1.2

Divide $\frac{d y}{d x}$ by $1$ .

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

Step 1.1.3

Simplify the right side.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Simplify the denominator.

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Step 1.1.3.1.1.1

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

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

Step 1.1.3.1.1.2

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 $b = x$ .

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

Step 1.1.3.1.1.3

Simplify.

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Step 1.1.3.1.1.3.1

One to any power is one.

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

Step 1.1.3.1.1.3.2

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

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

Step 1.1.3.1.2

Multiply $\frac{x^{2} y}{\sqrt{(1 + x) (1 - x + x^{2})}}$ by $\frac{\sqrt{(1 + x) (1 - x + x^{2})}}{\sqrt{(1 + x) (1 - x + x^{2})}}$ .

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

Step 1.1.3.1.3

Combine and simplify the denominator.

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Step 1.1.3.1.3.1

Multiply $\frac{x^{2} y}{\sqrt{(1 + x) (1 - x + x^{2})}}$ by $\frac{\sqrt{(1 + x) (1 - x + x^{2})}}{\sqrt{(1 + x) (1 - x + x^{2})}}$ .

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

Step 1.1.3.1.3.2

Raise $\sqrt{(1 + x) (1 - x + x^{2})}$ to the power of $1$ .

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

Step 1.1.3.1.3.3

Raise $\sqrt{(1 + x) (1 - x + x^{2})}$ to the power of $1$ .

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

Step 1.1.3.1.3.4

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

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

Step 1.1.3.1.3.5

Add $1$ and $1$ .

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

Step 1.1.3.1.3.6

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

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Step 1.1.3.1.3.6.1

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

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

Step 1.1.3.1.3.6.2

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

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

Step 1.1.3.1.3.6.3

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

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

Step 1.1.3.1.3.6.4

Cancel the common factor of $2$ .

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Step 1.1.3.1.3.6.4.1

Cancel the common factor.

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

Step 1.1.3.1.3.6.4.2

Rewrite the expression.

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

Step 1.1.3.1.3.6.5

Simplify.

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

Step 1.1.3.1.4

Simplify the denominator.

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Step 1.1.3.1.4.1

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

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

Step 1.1.3.1.4.2

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 $b = x$ .

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

Step 1.1.3.1.4.3

Simplify.

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Step 1.1.3.1.4.3.1

One to any power is one.

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

Step 1.1.3.1.4.3.2

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

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

Step 1.1.3.1.5

Multiply $\frac{x^{2}}{\sqrt{(1 + x) (1 - x + x^{2})}}$ by $\frac{\sqrt{(1 + x) (1 - x + x^{2})}}{\sqrt{(1 + x) (1 - x + x^{2})}}$ .

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

Step 1.1.3.1.6

Combine and simplify the denominator.

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Step 1.1.3.1.6.1

Multiply $\frac{x^{2}}{\sqrt{(1 + x) (1 - x + x^{2})}}$ by $\frac{\sqrt{(1 + x) (1 - x + x^{2})}}{\sqrt{(1 + x) (1 - x + x^{2})}}$ .

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

Step 1.1.3.1.6.2

Raise $\sqrt{(1 + x) (1 - x + x^{2})}$ to the power of $1$ .

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

Step 1.1.3.1.6.3

Raise $\sqrt{(1 + x) (1 - x + x^{2})}$ to the power of $1$ .

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

Step 1.1.3.1.6.4

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

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

Step 1.1.3.1.6.5

Add $1$ and $1$ .

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

Step 1.1.3.1.6.6

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

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Step 1.1.3.1.6.6.1

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

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

Step 1.1.3.1.6.6.2

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

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

Step 1.1.3.1.6.6.3

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

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

Step 1.1.3.1.6.6.4

Cancel the common factor of $2$ .

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Step 1.1.3.1.6.6.4.1

Cancel the common factor.

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

Step 1.1.3.1.6.6.4.2

Rewrite the expression.

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

Step 1.1.3.1.6.6.5

Simplify.

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

Step 1.1.3.2

Simplify terms.

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Step 1.1.3.2.1

Combine the numerators over the common denominator.

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

Step 1.1.3.2.2

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

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Step 1.1.3.2.2.1

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

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

Step 1.1.3.2.2.2

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

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

Step 1.1.3.2.2.3

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

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

Step 1.2

Regroup factors.

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

Step 1.3

Multiply both sides by $\frac{1}{y + 1}$ .

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

Step 1.4

Cancel the common factor of $y + 1$ .

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Step 1.4.1

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

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

Step 1.4.2

Cancel the common factor.

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

Step 1.4.3

Rewrite the expression.

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

Step 1.5

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Let $u_{1} = y + 1$ . Then $d u_{1} = d y$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 2.2.1.1

Let $u_{1} = y + 1$ . Find $\frac{d u_{1}}{d y}$ .

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Step 2.2.1.1.1

Differentiate $y + 1$ .

$\frac{d}{d y} [y + 1]$

Step 2.2.1.1.2

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

$\frac{d}{d y} [y] + \frac{d}{d y} [1]$

Step 2.2.1.1.3

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

$1 + \frac{d}{d y} [1]$

Step 2.2.1.1.4

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

$1 + 0$

Step 2.2.1.1.5

Add $1$ and $0$ .

$1$

Step 2.2.1.2

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

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

Step 2.2.2

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

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

Step 2.2.3

Replace all occurrences of $u_{1}$ with $y + 1$ .

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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Step 2.3.1.1

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

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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 $u_{2}$ and $d u_{2}$ .

$\ln (| y + 1 |) + C_{1} = \int \frac{\sqrt{u_{2}}}{u_{2}} \cdot \frac{1}{3} d u_{2}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $\frac{\sqrt{u_{2}}}{u_{2}}$ by $\frac{1}{3}$ .

$\ln (| y + 1 |) + C_{1} = \int \frac{\sqrt{u_{2}}}{u_{2} \cdot 3} d u_{2}$

Step 2.3.2.2

Move $3$ to the left of $u_{2}$ .

$\ln (| y + 1 |) + C_{1} = \int \frac{\sqrt{u_{2}}}{3 u_{2}} d u_{2}$

Step 2.3.3

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

$\ln (| y + 1 |) + C_{1} = \frac{1}{3} \int \frac{\sqrt{u_{2}}}{u_{2}} d u_{2}$

Step 2.3.4

Simplify the expression.

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Step 2.3.4.1

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

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

Step 2.3.4.2

Simplify.

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Step 2.3.4.2.1

Move ${u_{2}}^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 2.3.4.2.2

Multiply $u_{2}$ by ${u_{2}}^{- \frac{1}{2}}$ by adding the exponents.

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Step 2.3.4.2.2.1

Multiply $u_{2}$ by ${u_{2}}^{- \frac{1}{2}}$ .

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Step 2.3.4.2.2.1.1

Raise $u_{2}$ to the power of $1$ .

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

Step 2.3.4.2.2.1.2

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

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

Step 2.3.4.2.2.2

Write $1$ as a fraction with a common denominator.

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

Step 2.3.4.2.2.3

Combine the numerators over the common denominator.

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

Step 2.3.4.2.2.4

Subtract $1$ from $2$ .

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

Step 2.3.4.3

Apply basic rules of exponents.

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Step 2.3.4.3.1

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

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

Step 2.3.4.3.2

Multiply the exponents in ${({u_{2}}^{\frac{1}{2}})}^{- 1}$ .

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Step 2.3.4.3.2.1

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

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

Step 2.3.4.3.2.2

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

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

Step 2.3.4.3.2.3

Move the negative in front of the fraction.

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

Step 2.3.5

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

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

Step 2.3.6

Simplify.

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Step 2.3.6.1

Rewrite $\frac{1}{3} (2 {u_{2}}^{\frac{1}{2}} + C_{2})$ as $\frac{1}{3} \cdot 2 {u_{2}}^{\frac{1}{2}} + C_{2}$ .

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

Step 2.3.6.2

Combine $\frac{1}{3}$ and $2$ .

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

Step 2.3.7

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

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Solve for $y$ .

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Step 3.3.1

Rewrite the equation as $| y + 1 | = e^{\frac{2}{3} \cdot {((1 + x) (1 - x + x^{2}))}^{\frac{1}{2}} + C}$ .

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

Step 3.3.2

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

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Step 3.3.2.1

Simplify each term.

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Step 3.3.2.1.1

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

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

Step 3.3.2.1.2

Simplify each term.

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Step 3.3.2.1.2.1

Multiply $1$ by $1$ .

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

Step 3.3.2.1.2.2

Multiply $- x$ by $1$ .

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

Step 3.3.2.1.2.3

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

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

Step 3.3.2.1.2.4

Multiply $x$ by $1$ .

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

Step 3.3.2.1.2.5

Rewrite using the commutative property of multiplication.

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

Step 3.3.2.1.2.6

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

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Step 3.3.2.1.2.6.1

Move $x$ .

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

Step 3.3.2.1.2.6.2

Multiply $x$ by $x$ .

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

Step 3.3.2.1.2.7

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

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Step 3.3.2.1.2.7.1

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

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Step 3.3.2.1.2.7.1.1

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

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

Step 3.3.2.1.2.7.1.2

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

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

Step 3.3.2.1.2.7.2

Add $1$ and $2$ .

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

Step 3.3.2.1.3

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

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Step 3.3.2.1.3.1

Add $- x$ and $x$ .

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

Step 3.3.2.1.3.2

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

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

Step 3.3.2.1.3.3

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

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

Step 3.3.2.1.3.4

Add $1$ and $0$ .

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

Step 3.3.2.1.4

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

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

Step 3.3.2.2

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

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

Step 3.3.2.3

Simplify terms.

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Step 3.3.2.3.1

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

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

Step 3.3.2.3.2

Combine the numerators over the common denominator.

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

Step 3.3.2.4

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

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

Step 3.3.3

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

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

Step 3.3.4

Move all terms not containing $y$ to the right side of the equation.

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Step 3.3.4.1

Subtract $1$ from both sides of the equation.

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

Step 3.3.4.2

Simplify each term.

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Step 3.3.4.2.1

Split the fraction $\frac{2 {(1 + x^{3})}^{\frac{1}{2}} + 3 C}{3}$ into two fractions.

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

Step 3.3.4.2.2

Cancel the common factor of $3$ .

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Step 3.3.4.2.2.1

Cancel the common factor.

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

Step 3.3.4.2.2.2

Divide $C$ by $1$ .

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

Step 4

Group the constant terms together.

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Step 4.1

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

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

Step 4.2

Reorder $e^{\frac{2 {(1 + x^{3})}^{\frac{1}{2}}}{3}}$ and $e^{C}$ .

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

Step 4.3

Combine constants with the plus or minus.

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