Calculus Examples

$(8 + x^{12}) \frac{d y}{d x} = \frac{x^{11}}{y}$

Step 1

Separate the variables.

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Step 1.1

Divide each term in $(8 + x^{12}) \frac{d y}{d x} = \frac{x^{11}}{y}$ by $8 + x^{12}$ and simplify.

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Step 1.1.1

Divide each term in $(8 + x^{12}) \frac{d y}{d x} = \frac{x^{11}}{y}$ by $8 + x^{12}$ .

$\frac{(8 + x^{12}) \frac{d y}{d x}}{8 + x^{12}} = \frac{\frac{x^{11}}{y}}{8 + x^{12}}$

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of $8 + x^{12}$ .

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Step 1.1.2.1.1

Cancel the common factor.

$\frac{(8 + x^{12}) \frac{d y}{d x}}{8 + x^{12}} = \frac{\frac{x^{11}}{y}}{8 + x^{12}}$

Step 1.1.2.1.2

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

$\frac{d y}{d x} = \frac{\frac{x^{11}}{y}}{8 + x^{12}}$

Step 1.1.3

Simplify the right side.

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Step 1.1.3.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{d y}{d x} = \frac{x^{11}}{y} \cdot \frac{1}{8 + x^{12}}$

Step 1.1.3.2

Simplify the denominator.

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Step 1.1.3.2.1

Rewrite $8$ as $2^{3}$ .

$\frac{d y}{d x} = \frac{x^{11}}{y} \cdot \frac{1}{2^{3} + x^{12}}$

Step 1.1.3.2.2

Rewrite $x^{12}$ as ${(x^{4})}^{3}$ .

$\frac{d y}{d x} = \frac{x^{11}}{y} \cdot \frac{1}{2^{3} + {(x^{4})}^{3}}$

Step 1.1.3.2.3

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 $a = 2$ and $b = x^{4}$ .

$\frac{d y}{d x} = \frac{x^{11}}{y} \cdot \frac{1}{(2 + x^{4}) (2^{2} - 2 x^{4} + {(x^{4})}^{2})}$

Step 1.1.3.2.4

Simplify.

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Step 1.1.3.2.4.1

Raise $2$ to the power of $2$ .

$\frac{d y}{d x} = \frac{x^{11}}{y} \cdot \frac{1}{(2 + x^{4}) (4 - 2 x^{4} + {(x^{4})}^{2})}$

Step 1.1.3.2.4.2

Multiply the exponents in ${(x^{4})}^{2}$ .

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Step 1.1.3.2.4.2.1

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

$\frac{d y}{d x} = \frac{x^{11}}{y} \cdot \frac{1}{(2 + x^{4}) (4 - 2 x^{4} + x^{4 \cdot 2})}$

Step 1.1.3.2.4.2.2

Multiply $4$ by $2$ .

$\frac{d y}{d x} = \frac{x^{11}}{y} \cdot \frac{1}{(2 + x^{4}) (4 - 2 x^{4} + x^{8})}$

Step 1.1.3.3

Combine fractions.

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Step 1.1.3.3.1

Combine.

$\frac{d y}{d x} = \frac{x^{11} \cdot 1}{y ((2 + x^{4}) (4 - 2 x^{4} + x^{8}))}$

Step 1.1.3.3.2

Multiply $x^{11}$ by $1$ .

$\frac{d y}{d x} = \frac{x^{11}}{y (2 + x^{4}) (4 - 2 x^{4} + x^{8})}$

Step 1.2

Regroup factors.

$\frac{d y}{d x} = \frac{x^{11}}{(2 + x^{4}) (4 - 2 x^{4} + x^{8})} \cdot \frac{1}{y}$

Step 1.3

Multiply both sides by $y$ .

$y \frac{d y}{d x} = y (\frac{x^{11}}{(2 + x^{4}) (4 - 2 x^{4} + x^{8})} \cdot \frac{1}{y})$

Step 1.4

Simplify.

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Step 1.4.1

Combine.

$y \frac{d y}{d x} = y \frac{x^{11} \cdot 1}{(2 + x^{4}) (4 - 2 x^{4} + x^{8}) y}$

Step 1.4.2

Cancel the common factor of $y$ .

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Step 1.4.2.1

Factor $y$ out of $(2 + x^{4}) (4 - 2 x^{4} + x^{8}) y$ .

$y \frac{d y}{d x} = y \frac{x^{11} \cdot 1}{y ((2 + x^{4}) (4 - 2 x^{4} + x^{8}))}$

Step 1.4.2.2

Cancel the common factor.

$y \frac{d y}{d x} = y \frac{x^{11} \cdot 1}{y ((2 + x^{4}) (4 - 2 x^{4} + x^{8}))}$

Step 1.4.2.3

Rewrite the expression.

$y \frac{d y}{d x} = \frac{x^{11} \cdot 1}{(2 + x^{4}) (4 - 2 x^{4} + x^{8})}$

Step 1.4.3

Multiply $x^{11}$ by $1$ .

$y \frac{d y}{d x} = \frac{x^{11}}{(2 + x^{4}) (4 - 2 x^{4} + x^{8})}$

Step 1.5

Rewrite the equation.

$y d y = \frac{x^{11}}{(2 + x^{4}) (4 - 2 x^{4} + x^{8})} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y d y = \int \frac{x^{11}}{(2 + x^{4}) (4 - 2 x^{4} + x^{8})} d x$

Step 2.2

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

$\frac{1}{2} y^{2} + C_{1} = \int \frac{x^{11}}{(2 + x^{4}) (4 - 2 x^{4} + x^{8})} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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Step 2.3.1.1

Let $u_{1} = x^{2}$ . Find $\frac{d u_{1}}{d x}$ .

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Step 2.3.1.1.1

Differentiate $x^{2}$ .

$\frac{d}{d x} [x^{2}]$

Step 2.3.1.1.2

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

$2 x$

Step 2.3.1.2

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

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{\sqrt{u_{1}}}^{10}}{(2 + {\sqrt{u_{1}}}^{4}) (4 - 2 {\sqrt{u_{1}}}^{4} + {\sqrt{u_{1}}}^{8})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Rewrite ${\sqrt{u_{1}}}^{10}$ as ${u_{1}}^{5}$ .

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Step 2.3.2.1.1

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

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{({u_{1}}^{\frac{1}{2}})}^{10}}{(2 + {\sqrt{u_{1}}}^{4}) (4 - 2 {\sqrt{u_{1}}}^{4} + {\sqrt{u_{1}}}^{8})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.2.1.2

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

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{u_{1}}^{\frac{1}{2} \cdot 10}}{(2 + {\sqrt{u_{1}}}^{4}) (4 - 2 {\sqrt{u_{1}}}^{4} + {\sqrt{u_{1}}}^{8})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.2.1.3

Combine $\frac{1}{2}$ and $10$ .

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{u_{1}}^{\frac{10}{2}}}{(2 + {\sqrt{u_{1}}}^{4}) (4 - 2 {\sqrt{u_{1}}}^{4} + {\sqrt{u_{1}}}^{8})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.2.1.4

Cancel the common factor of $10$ and $2$ .

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Step 2.3.2.1.4.1

Factor $2$ out of $10$ .

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{u_{1}}^{\frac{2 \cdot 5}{2}}}{(2 + {\sqrt{u_{1}}}^{4}) (4 - 2 {\sqrt{u_{1}}}^{4} + {\sqrt{u_{1}}}^{8})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.2.1.4.2

Cancel the common factors.

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Step 2.3.2.1.4.2.1

Factor $2$ out of $2$ .

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{u_{1}}^{\frac{2 \cdot 5}{2 (1)}}}{(2 + {\sqrt{u_{1}}}^{4}) (4 - 2 {\sqrt{u_{1}}}^{4} + {\sqrt{u_{1}}}^{8})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.2.1.4.2.2

Cancel the common factor.

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{u_{1}}^{\frac{2 \cdot 5}{2 \cdot 1}}}{(2 + {\sqrt{u_{1}}}^{4}) (4 - 2 {\sqrt{u_{1}}}^{4} + {\sqrt{u_{1}}}^{8})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.2.1.4.2.3

Rewrite the expression.

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{u_{1}}^{\frac{5}{1}}}{(2 + {\sqrt{u_{1}}}^{4}) (4 - 2 {\sqrt{u_{1}}}^{4} + {\sqrt{u_{1}}}^{8})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.2.1.4.2.4

Divide $5$ by $1$ .

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{u_{1}}^{5}}{(2 + {\sqrt{u_{1}}}^{4}) (4 - 2 {\sqrt{u_{1}}}^{4} + {\sqrt{u_{1}}}^{8})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.2.2

Rewrite ${\sqrt{u_{1}}}^{4}$ as ${u_{1}}^{2}$ .

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Step 2.3.2.2.1

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

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{u_{1}}^{5}}{(2 + {({u_{1}}^{\frac{1}{2}})}^{4}) (4 - 2 {\sqrt{u_{1}}}^{4} + {\sqrt{u_{1}}}^{8})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.2.2.2

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

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{u_{1}}^{5}}{(2 + {u_{1}}^{\frac{1}{2} \cdot 4}) (4 - 2 {\sqrt{u_{1}}}^{4} + {\sqrt{u_{1}}}^{8})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.2.2.3

Combine $\frac{1}{2}$ and $4$ .

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{u_{1}}^{5}}{(2 + {u_{1}}^{\frac{4}{2}}) (4 - 2 {\sqrt{u_{1}}}^{4} + {\sqrt{u_{1}}}^{8})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.2.2.4

Cancel the common factor of $4$ and $2$ .

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Step 2.3.2.2.4.1

Factor $2$ out of $4$ .

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{u_{1}}^{5}}{(2 + {u_{1}}^{\frac{2 \cdot 2}{2}}) (4 - 2 {\sqrt{u_{1}}}^{4} + {\sqrt{u_{1}}}^{8})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.2.2.4.2

Cancel the common factors.

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Step 2.3.2.2.4.2.1

Factor $2$ out of $2$ .

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{u_{1}}^{5}}{(2 + {u_{1}}^{\frac{2 \cdot 2}{2 (1)}}) (4 - 2 {\sqrt{u_{1}}}^{4} + {\sqrt{u_{1}}}^{8})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.2.2.4.2.2

Cancel the common factor.

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{u_{1}}^{5}}{(2 + {u_{1}}^{\frac{2 \cdot 2}{2 \cdot 1}}) (4 - 2 {\sqrt{u_{1}}}^{4} + {\sqrt{u_{1}}}^{8})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.2.2.4.2.3

Rewrite the expression.

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{u_{1}}^{5}}{(2 + {u_{1}}^{\frac{2}{1}}) (4 - 2 {\sqrt{u_{1}}}^{4} + {\sqrt{u_{1}}}^{8})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.2.2.4.2.4

Divide $2$ by $1$ .

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{u_{1}}^{5}}{(2 + {u_{1}}^{2}) (4 - 2 {\sqrt{u_{1}}}^{4} + {\sqrt{u_{1}}}^{8})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.2.3

Rewrite ${\sqrt{u_{1}}}^{4}$ as ${u_{1}}^{2}$ .

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Step 2.3.2.3.1

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

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{u_{1}}^{5}}{(2 + {u_{1}}^{2}) (4 - 2 {({u_{1}}^{\frac{1}{2}})}^{4} + {\sqrt{u_{1}}}^{8})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.2.3.2

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

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{u_{1}}^{5}}{(2 + {u_{1}}^{2}) (4 - 2 {u_{1}}^{\frac{1}{2} \cdot 4} + {\sqrt{u_{1}}}^{8})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.2.3.3

Combine $\frac{1}{2}$ and $4$ .

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{u_{1}}^{5}}{(2 + {u_{1}}^{2}) (4 - 2 {u_{1}}^{\frac{4}{2}} + {\sqrt{u_{1}}}^{8})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.2.3.4

Cancel the common factor of $4$ and $2$ .

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Step 2.3.2.3.4.1

Factor $2$ out of $4$ .

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{u_{1}}^{5}}{(2 + {u_{1}}^{2}) (4 - 2 {u_{1}}^{\frac{2 \cdot 2}{2}} + {\sqrt{u_{1}}}^{8})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.2.3.4.2

Cancel the common factors.

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Step 2.3.2.3.4.2.1

Factor $2$ out of $2$ .

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{u_{1}}^{5}}{(2 + {u_{1}}^{2}) (4 - 2 {u_{1}}^{\frac{2 \cdot 2}{2 (1)}} + {\sqrt{u_{1}}}^{8})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.2.3.4.2.2

Cancel the common factor.

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{u_{1}}^{5}}{(2 + {u_{1}}^{2}) (4 - 2 {u_{1}}^{\frac{2 \cdot 2}{2 \cdot 1}} + {\sqrt{u_{1}}}^{8})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.2.3.4.2.3

Rewrite the expression.

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{u_{1}}^{5}}{(2 + {u_{1}}^{2}) (4 - 2 {u_{1}}^{\frac{2}{1}} + {\sqrt{u_{1}}}^{8})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.2.3.4.2.4

Divide $2$ by $1$ .

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{u_{1}}^{5}}{(2 + {u_{1}}^{2}) (4 - 2 {u_{1}}^{2} + {\sqrt{u_{1}}}^{8})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.2.4

Rewrite ${\sqrt{u_{1}}}^{8}$ as ${u_{1}}^{4}$ .

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Step 2.3.2.4.1

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

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{u_{1}}^{5}}{(2 + {u_{1}}^{2}) (4 - 2 {u_{1}}^{2} + {({u_{1}}^{\frac{1}{2}})}^{8})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.2.4.2

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

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{u_{1}}^{5}}{(2 + {u_{1}}^{2}) (4 - 2 {u_{1}}^{2} + {u_{1}}^{\frac{1}{2} \cdot 8})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.2.4.3

Combine $\frac{1}{2}$ and $8$ .

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{u_{1}}^{5}}{(2 + {u_{1}}^{2}) (4 - 2 {u_{1}}^{2} + {u_{1}}^{\frac{8}{2}})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.2.4.4

Cancel the common factor of $8$ and $2$ .

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Step 2.3.2.4.4.1

Factor $2$ out of $8$ .

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{u_{1}}^{5}}{(2 + {u_{1}}^{2}) (4 - 2 {u_{1}}^{2} + {u_{1}}^{\frac{2 \cdot 4}{2}})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.2.4.4.2

Cancel the common factors.

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Step 2.3.2.4.4.2.1

Factor $2$ out of $2$ .

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{u_{1}}^{5}}{(2 + {u_{1}}^{2}) (4 - 2 {u_{1}}^{2} + {u_{1}}^{\frac{2 \cdot 4}{2 (1)}})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.2.4.4.2.2

Cancel the common factor.

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{u_{1}}^{5}}{(2 + {u_{1}}^{2}) (4 - 2 {u_{1}}^{2} + {u_{1}}^{\frac{2 \cdot 4}{2 \cdot 1}})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.2.4.4.2.3

Rewrite the expression.

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{u_{1}}^{5}}{(2 + {u_{1}}^{2}) (4 - 2 {u_{1}}^{2} + {u_{1}}^{\frac{4}{1}})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.2.4.4.2.4

Divide $4$ by $1$ .

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{u_{1}}^{5}}{(2 + {u_{1}}^{2}) (4 - 2 {u_{1}}^{2} + {u_{1}}^{4})} \cdot \frac{1}{2} d u_{1}$

Step 2.3.2.5

Multiply $\frac{{u_{1}}^{5}}{(2 + {u_{1}}^{2}) (4 - 2 {u_{1}}^{2} + {u_{1}}^{4})}$ by $\frac{1}{2}$ .

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{u_{1}}^{5}}{(2 + {u_{1}}^{2}) (4 - 2 {u_{1}}^{2} + {u_{1}}^{4}) \cdot 2} d u_{1}$

Step 2.3.2.6

Move $2$ to the left of $(2 + {u_{1}}^{2}) (4 - 2 {u_{1}}^{2} + {u_{1}}^{4})$ .

$\frac{1}{2} y^{2} + C_{1} = \int \frac{{u_{1}}^{5}}{2 (2 + {u_{1}}^{2}) (4 - 2 {u_{1}}^{2} + {u_{1}}^{4})} d u_{1}$

Step 2.3.3

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

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{2} \int \frac{{u_{1}}^{5}}{(2 + {u_{1}}^{2}) (4 - 2 {u_{1}}^{2} + {u_{1}}^{4})} d u_{1}$

Step 2.3.4

Let $u_{2} = {u_{1}}^{2}$ . Then $d u_{2} = 2 u_{1} d u_{1}$ , so $\frac{1}{2} d u_{2} = u_{1} d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 2.3.4.1

Let $u_{2} = {u_{1}}^{2}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 2.3.4.1.1

Differentiate ${u_{1}}^{2}$ .

$\frac{d}{d u_{1}} [{u_{1}}^{2}]$

Step 2.3.4.1.2

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

$2 u_{1}$

Step 2.3.4.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{2} \int \frac{{\sqrt{u_{2}}}^{4}}{(2 + u_{2}) (4 - 2 u_{2} + {\sqrt{u_{2}}}^{4})} \cdot \frac{1}{2} d u_{2}$

Step 2.3.5

Simplify.

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Step 2.3.5.1

Rewrite ${\sqrt{u_{2}}}^{4}$ as ${u_{2}}^{2}$ .

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Step 2.3.5.1.1

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

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{2} \int \frac{{({u_{2}}^{\frac{1}{2}})}^{4}}{(2 + u_{2}) (4 - 2 u_{2} + {\sqrt{u_{2}}}^{4})} \cdot \frac{1}{2} d u_{2}$

Step 2.3.5.1.2

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

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{2} \int \frac{{u_{2}}^{\frac{1}{2} \cdot 4}}{(2 + u_{2}) (4 - 2 u_{2} + {\sqrt{u_{2}}}^{4})} \cdot \frac{1}{2} d u_{2}$

Step 2.3.5.1.3

Combine $\frac{1}{2}$ and $4$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{2} \int \frac{{u_{2}}^{\frac{4}{2}}}{(2 + u_{2}) (4 - 2 u_{2} + {\sqrt{u_{2}}}^{4})} \cdot \frac{1}{2} d u_{2}$

Step 2.3.5.1.4

Cancel the common factor of $4$ and $2$ .

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Step 2.3.5.1.4.1

Factor $2$ out of $4$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{2} \int \frac{{u_{2}}^{\frac{2 \cdot 2}{2}}}{(2 + u_{2}) (4 - 2 u_{2} + {\sqrt{u_{2}}}^{4})} \cdot \frac{1}{2} d u_{2}$

Step 2.3.5.1.4.2

Cancel the common factors.

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Step 2.3.5.1.4.2.1

Factor $2$ out of $2$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{2} \int \frac{{u_{2}}^{\frac{2 \cdot 2}{2 (1)}}}{(2 + u_{2}) (4 - 2 u_{2} + {\sqrt{u_{2}}}^{4})} \cdot \frac{1}{2} d u_{2}$

Step 2.3.5.1.4.2.2

Cancel the common factor.

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{2} \int \frac{{u_{2}}^{\frac{2 \cdot 2}{2 \cdot 1}}}{(2 + u_{2}) (4 - 2 u_{2} + {\sqrt{u_{2}}}^{4})} \cdot \frac{1}{2} d u_{2}$

Step 2.3.5.1.4.2.3

Rewrite the expression.

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{2} \int \frac{{u_{2}}^{\frac{2}{1}}}{(2 + u_{2}) (4 - 2 u_{2} + {\sqrt{u_{2}}}^{4})} \cdot \frac{1}{2} d u_{2}$

Step 2.3.5.1.4.2.4

Divide $2$ by $1$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{2} \int \frac{{u_{2}}^{2}}{(2 + u_{2}) (4 - 2 u_{2} + {\sqrt{u_{2}}}^{4})} \cdot \frac{1}{2} d u_{2}$

Step 2.3.5.2

Rewrite ${\sqrt{u_{2}}}^{4}$ as ${u_{2}}^{2}$ .

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Step 2.3.5.2.1

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

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{2} \int \frac{{u_{2}}^{2}}{(2 + u_{2}) (4 - 2 u_{2} + {({u_{2}}^{\frac{1}{2}})}^{4})} \cdot \frac{1}{2} d u_{2}$

Step 2.3.5.2.2

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

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{2} \int \frac{{u_{2}}^{2}}{(2 + u_{2}) (4 - 2 u_{2} + {u_{2}}^{\frac{1}{2} \cdot 4})} \cdot \frac{1}{2} d u_{2}$

Step 2.3.5.2.3

Combine $\frac{1}{2}$ and $4$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{2} \int \frac{{u_{2}}^{2}}{(2 + u_{2}) (4 - 2 u_{2} + {u_{2}}^{\frac{4}{2}})} \cdot \frac{1}{2} d u_{2}$

Step 2.3.5.2.4

Cancel the common factor of $4$ and $2$ .

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Step 2.3.5.2.4.1

Factor $2$ out of $4$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{2} \int \frac{{u_{2}}^{2}}{(2 + u_{2}) (4 - 2 u_{2} + {u_{2}}^{\frac{2 \cdot 2}{2}})} \cdot \frac{1}{2} d u_{2}$

Step 2.3.5.2.4.2

Cancel the common factors.

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Step 2.3.5.2.4.2.1

Factor $2$ out of $2$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{2} \int \frac{{u_{2}}^{2}}{(2 + u_{2}) (4 - 2 u_{2} + {u_{2}}^{\frac{2 \cdot 2}{2 (1)}})} \cdot \frac{1}{2} d u_{2}$

Step 2.3.5.2.4.2.2

Cancel the common factor.

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{2} \int \frac{{u_{2}}^{2}}{(2 + u_{2}) (4 - 2 u_{2} + {u_{2}}^{\frac{2 \cdot 2}{2 \cdot 1}})} \cdot \frac{1}{2} d u_{2}$

Step 2.3.5.2.4.2.3

Rewrite the expression.

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{2} \int \frac{{u_{2}}^{2}}{(2 + u_{2}) (4 - 2 u_{2} + {u_{2}}^{\frac{2}{1}})} \cdot \frac{1}{2} d u_{2}$

Step 2.3.5.2.4.2.4

Divide $2$ by $1$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{2} \int \frac{{u_{2}}^{2}}{(2 + u_{2}) (4 - 2 u_{2} + {u_{2}}^{2})} \cdot \frac{1}{2} d u_{2}$

Step 2.3.5.3

Multiply $\frac{{u_{2}}^{2}}{(2 + u_{2}) (4 - 2 u_{2} + {u_{2}}^{2})}$ by $\frac{1}{2}$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{2} \int \frac{{u_{2}}^{2}}{(2 + u_{2}) (4 - 2 u_{2} + {u_{2}}^{2}) \cdot 2} d u_{2}$

Step 2.3.5.4

Move $2$ to the left of $(2 + u_{2}) (4 - 2 u_{2} + {u_{2}}^{2})$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{2} \int \frac{{u_{2}}^{2}}{2 (2 + u_{2}) (4 - 2 u_{2} + {u_{2}}^{2})} d u_{2}$

Step 2.3.6

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

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{2} (\frac{1}{2} \int \frac{{u_{2}}^{2}}{(2 + u_{2}) (4 - 2 u_{2} + {u_{2}}^{2})} d u_{2})$

Step 2.3.7

Simplify.

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Step 2.3.7.1

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{2 \cdot 2} \int \frac{{u_{2}}^{2}}{(2 + u_{2}) (4 - 2 u_{2} + {u_{2}}^{2})} d u_{2}$

Step 2.3.7.2

Multiply $2$ by $2$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{4} \int \frac{{u_{2}}^{2}}{(2 + u_{2}) (4 - 2 u_{2} + {u_{2}}^{2})} d u_{2}$

Step 2.3.8

Let $u_{3} = (2 + u_{2}) (4 - 2 u_{2} + {u_{2}}^{2})$ . Then $d u_{3} = 3 {u_{2}}^{2} d u_{2}$ , so $\frac{1}{3} d u_{3} = {u_{2}}^{2} d u_{2}$ . Rewrite using $u_{3}$ and $d$ $u_{3}$ .

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Step 2.3.8.1

Let $u_{3} = (2 + u_{2}) (4 - 2 u_{2} + {u_{2}}^{2})$ . Find $\frac{d u_{3}}{d u_{2}}$ .

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Step 2.3.8.1.1

Differentiate $(2 + u_{2}) (4 - 2 u_{2} + {u_{2}}^{2})$ .

$\frac{d}{d u_{2}} [(2 + u_{2}) (4 - 2 u_{2} + {u_{2}}^{2})]$

Step 2.3.8.1.2

Differentiate using the Product Rule which states that $\frac{d}{d u_{2}} [f (u_{2}) g (u_{2})]$ is $f (u_{2}) \frac{d}{d u_{2}} [g (u_{2})] + g (u_{2}) \frac{d}{d u_{2}} [f (u_{2})]$ where $f (u_{2}) = 2 + u_{2}$ and $g (u_{2}) = 4 - 2 u_{2} + {u_{2}}^{2}$ .

$(2 + u_{2}) \frac{d}{d u_{2}} [4 - 2 u_{2} + {u_{2}}^{2}] + (4 - 2 u_{2} + {u_{2}}^{2}) \frac{d}{d u_{2}} [2 + u_{2}]$

Step 2.3.8.1.3

Differentiate.

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Step 2.3.8.1.3.1

By the Sum Rule, the derivative of $4 - 2 u_{2} + {u_{2}}^{2}$ with respect to $u_{2}$ is $\frac{d}{d u_{2}} [4] + \frac{d}{d u_{2}} [- 2 u_{2}] + \frac{d}{d u_{2}} [{u_{2}}^{2}]$ .

$(2 + u_{2}) (\frac{d}{d u_{2}} [4] + \frac{d}{d u_{2}} [- 2 u_{2}] + \frac{d}{d u_{2}} [{u_{2}}^{2}]) + (4 - 2 u_{2} + {u_{2}}^{2}) \frac{d}{d u_{2}} [2 + u_{2}]$

Step 2.3.8.1.3.2

Since $4$ is constant with respect to $u_{2}$ , the derivative of $4$ with respect to $u_{2}$ is $0$ .

$(2 + u_{2}) (0 + \frac{d}{d u_{2}} [- 2 u_{2}] + \frac{d}{d u_{2}} [{u_{2}}^{2}]) + (4 - 2 u_{2} + {u_{2}}^{2}) \frac{d}{d u_{2}} [2 + u_{2}]$

Step 2.3.8.1.3.3

Add $0$ and $\frac{d}{d u_{2}} [- 2 u_{2}]$ .

$(2 + u_{2}) (\frac{d}{d u_{2}} [- 2 u_{2}] + \frac{d}{d u_{2}} [{u_{2}}^{2}]) + (4 - 2 u_{2} + {u_{2}}^{2}) \frac{d}{d u_{2}} [2 + u_{2}]$

Step 2.3.8.1.3.4

Since $- 2$ is constant with respect to $u_{2}$ , the derivative of $- 2 u_{2}$ with respect to $u_{2}$ is $- 2 \frac{d}{d u_{2}} [u_{2}]$ .

$(2 + u_{2}) (- 2 \frac{d}{d u_{2}} [u_{2}] + \frac{d}{d u_{2}} [{u_{2}}^{2}]) + (4 - 2 u_{2} + {u_{2}}^{2}) \frac{d}{d u_{2}} [2 + u_{2}]$

Step 2.3.8.1.3.5

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

$(2 + u_{2}) (- 2 \cdot 1 + \frac{d}{d u_{2}} [{u_{2}}^{2}]) + (4 - 2 u_{2} + {u_{2}}^{2}) \frac{d}{d u_{2}} [2 + u_{2}]$

Step 2.3.8.1.3.6

Multiply $- 2$ by $1$ .

$(2 + u_{2}) (- 2 + \frac{d}{d u_{2}} [{u_{2}}^{2}]) + (4 - 2 u_{2} + {u_{2}}^{2}) \frac{d}{d u_{2}} [2 + u_{2}]$

Step 2.3.8.1.3.7

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

$(2 + u_{2}) (- 2 + 2 u_{2}) + (4 - 2 u_{2} + {u_{2}}^{2}) \frac{d}{d u_{2}} [2 + u_{2}]$

Step 2.3.8.1.3.8

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

$(2 + u_{2}) (- 2 + 2 u_{2}) + (4 - 2 u_{2} + {u_{2}}^{2}) (\frac{d}{d u_{2}} [2] + \frac{d}{d u_{2}} [u_{2}])$

Step 2.3.8.1.3.9

Since $2$ is constant with respect to $u_{2}$ , the derivative of $2$ with respect to $u_{2}$ is $0$ .

$(2 + u_{2}) (- 2 + 2 u_{2}) + (4 - 2 u_{2} + {u_{2}}^{2}) (0 + \frac{d}{d u_{2}} [u_{2}])$

Step 2.3.8.1.3.10

Add $0$ and $\frac{d}{d u_{2}} [u_{2}]$ .

$(2 + u_{2}) (- 2 + 2 u_{2}) + (4 - 2 u_{2} + {u_{2}}^{2}) \frac{d}{d u_{2}} [u_{2}]$

Step 2.3.8.1.3.11

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

$(2 + u_{2}) (- 2 + 2 u_{2}) + (4 - 2 u_{2} + {u_{2}}^{2}) \cdot 1$

Step 2.3.8.1.3.12

Multiply $4 - 2 u_{2} + {u_{2}}^{2}$ by $1$ .

$(2 + u_{2}) (- 2 + 2 u_{2}) + 4 - 2 u_{2} + {u_{2}}^{2}$

Step 2.3.8.1.4

Simplify.

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Step 2.3.8.1.4.1

Apply the distributive property.

$(2 + u_{2}) \cdot - 2 + (2 + u_{2}) (2 u_{2}) + 4 - 2 u_{2} + {u_{2}}^{2}$

Step 2.3.8.1.4.2

Apply the distributive property.

$2 \cdot - 2 + u_{2} \cdot - 2 + (2 + u_{2}) (2 u_{2}) + 4 - 2 u_{2} + {u_{2}}^{2}$

Step 2.3.8.1.4.3

Apply the distributive property.

$2 \cdot - 2 + u_{2} \cdot - 2 + 2 (2 u_{2}) + u_{2} (2 u_{2}) + 4 - 2 u_{2} + {u_{2}}^{2}$

Step 2.3.8.1.4.4

Combine terms.

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Step 2.3.8.1.4.4.1

Multiply $2$ by $- 2$ .

$- 4 + u_{2} \cdot - 2 + 2 (2 u_{2}) + u_{2} (2 u_{2}) + 4 - 2 u_{2} + {u_{2}}^{2}$

Step 2.3.8.1.4.4.2

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

$- 4 - 2 \cdot u_{2} + 2 (2 u_{2}) + u_{2} (2 u_{2}) + 4 - 2 u_{2} + {u_{2}}^{2}$

Step 2.3.8.1.4.4.3

Multiply $2$ by $2$ .

$- 4 - 2 u_{2} + 4 u_{2} + u_{2} (2 u_{2}) + 4 - 2 u_{2} + {u_{2}}^{2}$

Step 2.3.8.1.4.4.4

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

$- 4 - 2 u_{2} + 4 u_{2} + 2 ({u_{2}}^{1} u_{2}) + 4 - 2 u_{2} + {u_{2}}^{2}$

Step 2.3.8.1.4.4.5

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

$- 4 - 2 u_{2} + 4 u_{2} + 2 ({u_{2}}^{1} {u_{2}}^{1}) + 4 - 2 u_{2} + {u_{2}}^{2}$

Step 2.3.8.1.4.4.6

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

$- 4 - 2 u_{2} + 4 u_{2} + 2 {u_{2}}^{1 + 1} + 4 - 2 u_{2} + {u_{2}}^{2}$

Step 2.3.8.1.4.4.7

Add $1$ and $1$ .

$- 4 - 2 u_{2} + 4 u_{2} + 2 {u_{2}}^{2} + 4 - 2 u_{2} + {u_{2}}^{2}$

Step 2.3.8.1.4.4.8

Add $- 2 u_{2}$ and $4 u_{2}$ .

$- 4 + 2 u_{2} + 2 {u_{2}}^{2} + 4 - 2 u_{2} + {u_{2}}^{2}$

Step 2.3.8.1.4.4.9

Add $- 4$ and $4$ .

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

Step 2.3.8.1.4.4.10

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

$2 u_{2} + 2 {u_{2}}^{2} - 2 u_{2} + {u_{2}}^{2}$

Step 2.3.8.1.4.4.11

Subtract $2 u_{2}$ from $2 u_{2}$ .

$2 {u_{2}}^{2} + 0 + {u_{2}}^{2}$

Step 2.3.8.1.4.4.12

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

$2 {u_{2}}^{2} + {u_{2}}^{2}$

Step 2.3.8.1.4.4.13

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

$3 {u_{2}}^{2}$

Step 2.3.8.2

Rewrite the problem using $u_{3}$ and $d u_{3}$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{4} \int \frac{1}{u_{3}} \cdot \frac{1}{3} d u_{3}$

Step 2.3.9

Simplify.

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Step 2.3.9.1

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

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{4} \int \frac{1}{u_{3} \cdot 3} d u_{3}$

Step 2.3.9.2

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

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{4} \int \frac{1}{3 u_{3}} d u_{3}$

Step 2.3.10

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

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{4} (\frac{1}{3} \int \frac{1}{u_{3}} d u_{3})$

Step 2.3.11

Simplify.

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Step 2.3.11.1

Multiply $\frac{1}{3}$ by $\frac{1}{4}$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{3 \cdot 4} \int \frac{1}{u_{3}} d u_{3}$

Step 2.3.11.2

Multiply $3$ by $4$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{12} \int \frac{1}{u_{3}} d u_{3}$

Step 2.3.12

The integral of $\frac{1}{u_{3}}$ with respect to $u_{3}$ is $\ln (| u_{3} |)$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{12} (\ln (| u_{3} |) + C_{2})$

Step 2.3.13

Simplify.

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{12} \ln (| u_{3} |) + C_{2}$

Step 2.3.14

Substitute back in for each integration substitution variable.

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Step 2.3.14.1

Replace all occurrences of $u_{3}$ with $(2 + u_{2}) (4 - 2 u_{2} + {u_{2}}^{2})$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{12} \ln (| (2 + u_{2}) (4 - 2 u_{2} + {u_{2}}^{2}) |) + C_{2}$

Step 2.3.14.2

Replace all occurrences of $u_{2}$ with ${u_{1}}^{2}$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{12} \ln (| (2 + {u_{1}}^{2}) (4 - 2 {u_{1}}^{2} + {({u_{1}}^{2})}^{2}) |) + C_{2}$

Step 2.3.14.3

Replace all occurrences of $u_{1}$ with $x^{2}$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{12} \ln (| (2 + {(x^{2})}^{2}) (4 - 2 {(x^{2})}^{2} + {({(x^{2})}^{2})}^{2}) |) + C_{2}$

Step 2.4

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

$\frac{1}{2} y^{2} = \frac{1}{12} \ln (| (2 + {(x^{2})}^{2}) (4 - 2 {(x^{2})}^{2} + {({(x^{2})}^{2})}^{2}) |) + C$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} y^{2}) = 2 (\frac{1}{12} \ln (| (2 + {(x^{2})}^{2}) (4 - 2 {(x^{2})}^{2} + {({(x^{2})}^{2})}^{2}) |) + C)$

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 $y^{2}$ .

$2 \frac{y^{2}}{2} = 2 (\frac{1}{12} \ln (| (2 + {(x^{2})}^{2}) (4 - 2 {(x^{2})}^{2} + {({(x^{2})}^{2})}^{2}) |) + C)$

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.

$2 \frac{y^{2}}{2} = 2 (\frac{1}{12} \ln (| (2 + {(x^{2})}^{2}) (4 - 2 {(x^{2})}^{2} + {({(x^{2})}^{2})}^{2}) |) + C)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{2} = 2 (\frac{1}{12} \ln (| (2 + {(x^{2})}^{2}) (4 - 2 {(x^{2})}^{2} + {({(x^{2})}^{2})}^{2}) |) + C)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $2 (\frac{1}{12} \ln (| (2 + {(x^{2})}^{2}) (4 - 2 {(x^{2})}^{2} + {({(x^{2})}^{2})}^{2}) |) + C)$ .

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Step 3.2.2.1.1

Simplify each term.

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Step 3.2.2.1.1.1

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

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Step 3.2.2.1.1.1.1

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

$y^{2} = 2 (\frac{1}{12} \ln (| (2 + x^{2 \cdot 2}) (4 - 2 {(x^{2})}^{2} + {({(x^{2})}^{2})}^{2}) |) + C)$

Step 3.2.2.1.1.1.2

Multiply $2$ by $2$ .

$y^{2} = 2 (\frac{1}{12} \ln (| (2 + x^{4}) (4 - 2 {(x^{2})}^{2} + {({(x^{2})}^{2})}^{2}) |) + C)$

Step 3.2.2.1.1.2

Simplify each term.

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Step 3.2.2.1.1.2.1

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

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Step 3.2.2.1.1.2.1.1

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

$y^{2} = 2 (\frac{1}{12} \ln (| (2 + x^{4}) (4 - 2 x^{2 \cdot 2} + {({(x^{2})}^{2})}^{2}) |) + C)$

Step 3.2.2.1.1.2.1.2

Multiply $2$ by $2$ .

$y^{2} = 2 (\frac{1}{12} \ln (| (2 + x^{4}) (4 - 2 x^{4} + {({(x^{2})}^{2})}^{2}) |) + C)$

Step 3.2.2.1.1.2.2

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

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Step 3.2.2.1.1.2.2.1

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

$y^{2} = 2 (\frac{1}{12} \ln (| (2 + x^{4}) (4 - 2 x^{4} + {(x^{2})}^{2 \cdot 2}) |) + C)$

Step 3.2.2.1.1.2.2.2

Multiply $2$ by $2$ .

$y^{2} = 2 (\frac{1}{12} \ln (| (2 + x^{4}) (4 - 2 x^{4} + {(x^{2})}^{4}) |) + C)$

Step 3.2.2.1.1.2.3

Multiply the exponents in ${(x^{2})}^{4}$ .

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Step 3.2.2.1.1.2.3.1

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

$y^{2} = 2 (\frac{1}{12} \ln (| (2 + x^{4}) (4 - 2 x^{4} + x^{2 \cdot 4}) |) + C)$

Step 3.2.2.1.1.2.3.2

Multiply $2$ by $4$ .

$y^{2} = 2 (\frac{1}{12} \ln (| (2 + x^{4}) (4 - 2 x^{4} + x^{8}) |) + C)$

Step 3.2.2.1.1.3

Expand $(2 + x^{4}) (4 - 2 x^{4} + x^{8})$ by multiplying each term in the first expression by each term in the second expression.

$y^{2} = 2 (\frac{1}{12} \ln (| 2 \cdot 4 + 2 (- 2 x^{4}) + 2 x^{8} + x^{4} \cdot 4 + x^{4} (- 2 x^{4}) + x^{4} x^{8} |) + C)$

Step 3.2.2.1.1.4

Simplify each term.

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Step 3.2.2.1.1.4.1

Multiply $2$ by $4$ .

$y^{2} = 2 (\frac{1}{12} \ln (| 8 + 2 (- 2 x^{4}) + 2 x^{8} + x^{4} \cdot 4 + x^{4} (- 2 x^{4}) + x^{4} x^{8} |) + C)$

Step 3.2.2.1.1.4.2

Multiply $- 2$ by $2$ .

$y^{2} = 2 (\frac{1}{12} \ln (| 8 - 4 x^{4} + 2 x^{8} + x^{4} \cdot 4 + x^{4} (- 2 x^{4}) + x^{4} x^{8} |) + C)$

Step 3.2.2.1.1.4.3

Move $4$ to the left of $x^{4}$ .

$y^{2} = 2 (\frac{1}{12} \ln (| 8 - 4 x^{4} + 2 x^{8} + 4 \cdot x^{4} + x^{4} (- 2 x^{4}) + x^{4} x^{8} |) + C)$

Step 3.2.2.1.1.4.4

Rewrite using the commutative property of multiplication.

$y^{2} = 2 (\frac{1}{12} \ln (| 8 - 4 x^{4} + 2 x^{8} + 4 x^{4} - 2 x^{4} x^{4} + x^{4} x^{8} |) + C)$

Step 3.2.2.1.1.4.5

Multiply $x^{4}$ by $x^{4}$ by adding the exponents.

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Step 3.2.2.1.1.4.5.1

Move $x^{4}$ .

$y^{2} = 2 (\frac{1}{12} \ln (| 8 - 4 x^{4} + 2 x^{8} + 4 x^{4} - 2 (x^{4} x^{4}) + x^{4} x^{8} |) + C)$

Step 3.2.2.1.1.4.5.2

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

$y^{2} = 2 (\frac{1}{12} \ln (| 8 - 4 x^{4} + 2 x^{8} + 4 x^{4} - 2 x^{4 + 4} + x^{4} x^{8} |) + C)$

Step 3.2.2.1.1.4.5.3

Add $4$ and $4$ .

$y^{2} = 2 (\frac{1}{12} \ln (| 8 - 4 x^{4} + 2 x^{8} + 4 x^{4} - 2 x^{8} + x^{4} x^{8} |) + C)$

Step 3.2.2.1.1.4.6

Multiply $x^{4}$ by $x^{8}$ by adding the exponents.

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Step 3.2.2.1.1.4.6.1

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

$y^{2} = 2 (\frac{1}{12} \ln (| 8 - 4 x^{4} + 2 x^{8} + 4 x^{4} - 2 x^{8} + x^{4 + 8} |) + C)$

Step 3.2.2.1.1.4.6.2

Add $4$ and $8$ .

$y^{2} = 2 (\frac{1}{12} \ln (| 8 - 4 x^{4} + 2 x^{8} + 4 x^{4} - 2 x^{8} + x^{12} |) + C)$

Step 3.2.2.1.1.5

Combine the opposite terms in $8 - 4 x^{4} + 2 x^{8} + 4 x^{4} - 2 x^{8} + x^{12}$ .

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Step 3.2.2.1.1.5.1

Add $- 4 x^{4}$ and $4 x^{4}$ .

$y^{2} = 2 (\frac{1}{12} \ln (| 8 + 2 x^{8} + 0 - 2 x^{8} + x^{12} |) + C)$

Step 3.2.2.1.1.5.2

Add $8 + 2 x^{8}$ and $0$ .

$y^{2} = 2 (\frac{1}{12} \ln (| 8 + 2 x^{8} - 2 x^{8} + x^{12} |) + C)$

Step 3.2.2.1.1.5.3

Subtract $2 x^{8}$ from $2 x^{8}$ .

$y^{2} = 2 (\frac{1}{12} \ln (| 8 + 0 + x^{12} |) + C)$

Step 3.2.2.1.1.5.4

Add $8$ and $0$ .

$y^{2} = 2 (\frac{1}{12} \ln (| 8 + x^{12} |) + C)$

Step 3.2.2.1.1.6

Combine $\frac{1}{12}$ and $\ln (| 8 + x^{12} |)$ .

$y^{2} = 2 (\frac{\ln (| 8 + x^{12} |)}{12} + C)$

Step 3.2.2.1.2

To write $C$ as a fraction with a common denominator, multiply by $\frac{12}{12}$ .

$y^{2} = 2 (\frac{\ln (| 8 + x^{12} |)}{12} + C \cdot \frac{12}{12})$

Step 3.2.2.1.3

Simplify terms.

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Step 3.2.2.1.3.1

Combine $C$ and $\frac{12}{12}$ .

$y^{2} = 2 (\frac{\ln (| 8 + x^{12} |)}{12} + \frac{C \cdot 12}{12})$

Step 3.2.2.1.3.2

Combine the numerators over the common denominator.

$y^{2} = 2 \frac{\ln (| 8 + x^{12} |) + C \cdot 12}{12}$

Step 3.2.2.1.3.3

Cancel the common factor of $2$ .

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Step 3.2.2.1.3.3.1

Factor $2$ out of $12$ .

$y^{2} = 2 \frac{\ln (| 8 + x^{12} |) + C \cdot 12}{2 (6)}$

Step 3.2.2.1.3.3.2

Cancel the common factor.

$y^{2} = 2 \frac{\ln (| 8 + x^{12} |) + C \cdot 12}{2 \cdot 6}$

Step 3.2.2.1.3.3.3

Rewrite the expression.

$y^{2} = \frac{\ln (| 8 + x^{12} |) + C \cdot 12}{6}$

Step 3.2.2.1.4

Move $12$ to the left of $C$ .

$y^{2} = \frac{\ln (| 8 + x^{12} |) + 12 C}{6}$

Step 3.3

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

$y = \pm \sqrt{\frac{\ln (| 8 + x^{12} |) + 12 C}{6}}$

Step 3.4

Simplify $\pm \sqrt{\frac{\ln (| 8 + x^{12} |) + 12 C}{6}}$ .

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Step 3.4.1

Rewrite $\sqrt{\frac{\ln (| 8 + x^{12} |) + 12 C}{6}}$ as $\frac{\sqrt{\ln (| 8 + x^{12} |) + 12 C}}{\sqrt{6}}$ .

$y = \pm \frac{\sqrt{\ln (| 8 + x^{12} |) + 12 C}}{\sqrt{6}}$

Step 3.4.2

Multiply $\frac{\sqrt{\ln (| 8 + x^{12} |) + 12 C}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$y = \pm \frac{\sqrt{\ln (| 8 + x^{12} |) + 12 C}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Step 3.4.3

Combine and simplify the denominator.

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Step 3.4.3.1

Multiply $\frac{\sqrt{\ln (| 8 + x^{12} |) + 12 C}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$y = \pm \frac{\sqrt{\ln (| 8 + x^{12} |) + 12 C} \sqrt{6}}{\sqrt{6} \sqrt{6}}$

Step 3.4.3.2

Raise $\sqrt{6}$ to the power of $1$ .

$y = \pm \frac{\sqrt{\ln (| 8 + x^{12} |) + 12 C} \sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}}$

Step 3.4.3.3

Raise $\sqrt{6}$ to the power of $1$ .

$y = \pm \frac{\sqrt{\ln (| 8 + x^{12} |) + 12 C} \sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}}$

Step 3.4.3.4

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

$y = \pm \frac{\sqrt{\ln (| 8 + x^{12} |) + 12 C} \sqrt{6}}{{\sqrt{6}}^{1 + 1}}$

Step 3.4.3.5

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{\ln (| 8 + x^{12} |) + 12 C} \sqrt{6}}{{\sqrt{6}}^{2}}$

Step 3.4.3.6

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 3.4.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$y = \pm \frac{\sqrt{\ln (| 8 + x^{12} |) + 12 C} \sqrt{6}}{{(6^{\frac{1}{2}})}^{2}}$

Step 3.4.3.6.2

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

$y = \pm \frac{\sqrt{\ln (| 8 + x^{12} |) + 12 C} \sqrt{6}}{6^{\frac{1}{2} \cdot 2}}$

Step 3.4.3.6.3

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

$y = \pm \frac{\sqrt{\ln (| 8 + x^{12} |) + 12 C} \sqrt{6}}{6^{\frac{2}{2}}}$

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.

$y = \pm \frac{\sqrt{\ln (| 8 + x^{12} |) + 12 C} \sqrt{6}}{6^{\frac{2}{2}}}$

Step 3.4.3.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{\ln (| 8 + x^{12} |) + 12 C} \sqrt{6}}{6^{1}}$

Step 3.4.3.6.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{\ln (| 8 + x^{12} |) + 12 C} \sqrt{6}}{6}$

Step 3.4.4

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{(\ln (| 8 + x^{12} |) + 12 C) \cdot 6}}{6}$

Step 3.4.5

Reorder factors in $\pm \frac{\sqrt{(\ln (| 8 + x^{12} |) + 12 C) \cdot 6}}{6}$ .

$y = \pm \frac{\sqrt{6 (\ln (| 8 + x^{12} |) + 12 C)}}{6}$

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.

$y = \frac{\sqrt{6 (\ln (| 8 + x^{12} |) + 12 C)}}{6}$

Step 3.5.2

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

$y = - \frac{\sqrt{6 (\ln (| 8 + x^{12} |) + 12 C)}}{6}$

Step 3.5.3

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

$y = \frac{\sqrt{6 (\ln (| 8 + x^{12} |) + 12 C)}}{6}$

$y = - \frac{\sqrt{6 (\ln (| 8 + x^{12} |) + 12 C)}}{6}$

$y = \frac{\sqrt{6 (\ln (| 8 + x^{12} |) + 12 C)}}{6}$

$y = - \frac{\sqrt{6 (\ln (| 8 + x^{12} |) + 12 C)}}{6}$

$y = \frac{\sqrt{6 (\ln (| 8 + x^{12} |) + 12 C)}}{6}$

$y = - \frac{\sqrt{6 (\ln (| 8 + x^{12} |) + 12 C)}}{6}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt{6 (\ln (| 8 + x^{12} |) + C)}}{6}$

$y = - \frac{\sqrt{6 (\ln (| 8 + x^{12} |) + C)}}{6}$