Calculus Examples

$x \frac{d v}{d x} = \frac{1 - 4 v^{2}}{3 v}$

Step 1

Separate the variables.

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Step 1.1

Divide each term in $x \frac{d v}{d x} = \frac{1 - 4 v^{2}}{3 v}$ by $x$ and simplify.

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Step 1.1.1

Divide each term in $x \frac{d v}{d x} = \frac{1 - 4 v^{2}}{3 v}$ by $x$ .

$\frac{x \frac{d v}{d x}}{x} = \frac{\frac{1 - 4 v^{2}}{3 v}}{x}$

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of $x$ .

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Step 1.1.2.1.1

Cancel the common factor.

$\frac{x \frac{d v}{d x}}{x} = \frac{\frac{1 - 4 v^{2}}{3 v}}{x}$

Step 1.1.2.1.2

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

$\frac{d v}{d x} = \frac{\frac{1 - 4 v^{2}}{3 v}}{x}$

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.

$\frac{d v}{d x} = \frac{1 - 4 v^{2}}{3 v} \cdot \frac{1}{x}$

Step 1.1.3.2

Simplify the numerator.

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Step 1.1.3.2.1

Rewrite $1$ as $1^{2}$ .

$\frac{d v}{d x} = \frac{1^{2} - 4 v^{2}}{3 v} \cdot \frac{1}{x}$

Step 1.1.3.2.2

Rewrite $4 v^{2}$ as ${(2 v)}^{2}$ .

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

Step 1.1.3.2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = 2 v$ .

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

Step 1.1.3.2.4

Multiply $2$ by $- 1$ .

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

Step 1.1.3.3

Multiply $\frac{(1 + 2 v) (1 - 2 v)}{3 v}$ by $\frac{1}{x}$ .

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

Step 1.2

Regroup factors.

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

Step 1.3

Multiply both sides by $\frac{3 v}{(1 + 2 v) (1 - 2 v)}$ .

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

Step 1.4

Simplify.

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Step 1.4.1

Multiply $\frac{(1 + 2 v) (1 - 2 v)}{3 v}$ by $\frac{1}{x}$ .

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

Step 1.4.2

Cancel the common factor of $3 v$ .

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Step 1.4.2.1

Factor $3 v$ out of $3 v x$ .

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

Step 1.4.2.2

Cancel the common factor.

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

Step 1.4.2.3

Rewrite the expression.

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

Step 1.4.3

Cancel the common factor of $(1 + 2 v) (1 - 2 v)$ .

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Step 1.4.3.1

Cancel the common factor.

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

Step 1.4.3.2

Rewrite the expression.

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

Step 1.5

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

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

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

Step 2.2.2

Let $u = (1 + 2 v) (1 - 2 v)$ . Then $d u = - 8 v d v$ , so $- \frac{1}{8} d u = v d v$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.2.1

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

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Step 2.2.2.1.1

Differentiate $(1 + 2 v) (1 - 2 v)$ .

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

Step 2.2.2.1.2

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

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

Step 2.2.2.1.3

Differentiate.

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Step 2.2.2.1.3.1

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

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

Step 2.2.2.1.3.2

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

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

Step 2.2.2.1.3.3

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

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

Step 2.2.2.1.3.4

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

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

Step 2.2.2.1.3.5

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

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

Step 2.2.2.1.3.6

Simplify the expression.

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Step 2.2.2.1.3.6.1

Multiply $- 2$ by $1$ .

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

Step 2.2.2.1.3.6.2

Move $- 2$ to the left of $1 + 2 v$ .

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

Step 2.2.2.1.3.7

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

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

Step 2.2.2.1.3.8

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

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

Step 2.2.2.1.3.9

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

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

Step 2.2.2.1.3.10

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

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

Step 2.2.2.1.3.11

Move $2$ to the left of $1 - 2 v$ .

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

Step 2.2.2.1.3.12

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

$- 2 (1 + 2 v) + 2 (1 - 2 v) \cdot 1$

Step 2.2.2.1.3.13

Multiply $2$ by $1$ .

$- 2 (1 + 2 v) + 2 (1 - 2 v)$

Step 2.2.2.1.4

Simplify.

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Step 2.2.2.1.4.1

Apply the distributive property.

$- 2 \cdot 1 - 2 (2 v) + 2 (1 - 2 v)$

Step 2.2.2.1.4.2

Apply the distributive property.

$- 2 \cdot 1 - 2 (2 v) + 2 \cdot 1 + 2 (- 2 v)$

Step 2.2.2.1.4.3

Combine terms.

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Step 2.2.2.1.4.3.1

Multiply $- 2$ by $1$ .

$- 2 - 2 (2 v) + 2 \cdot 1 + 2 (- 2 v)$

Step 2.2.2.1.4.3.2

Multiply $2$ by $- 2$ .

$- 2 - 4 v + 2 \cdot 1 + 2 (- 2 v)$

Step 2.2.2.1.4.3.3

Multiply $2$ by $1$ .

$- 2 - 4 v + 2 + 2 (- 2 v)$

Step 2.2.2.1.4.3.4

Multiply $- 2$ by $2$ .

$- 2 - 4 v + 2 - 4 v$

Step 2.2.2.1.4.3.5

Add $- 2$ and $2$ .

$- 4 v + 0 - 4 v$

Step 2.2.2.1.4.3.6

Add $- 4 v$ and $0$ .

$- 4 v - 4 v$

Step 2.2.2.1.4.3.7

Subtract $4 v$ from $- 4 v$ .

$- 8 v$

Step 2.2.2.2

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

$3 \int \frac{1}{u} \cdot \frac{1}{- 8} d u = \int \frac{1}{x} d x$

Step 2.2.3

Simplify.

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Step 2.2.3.1

Move the negative in front of the fraction.

$3 \int \frac{1}{u} (- \frac{1}{8}) d u = \int \frac{1}{x} d x$

Step 2.2.3.2

Multiply $\frac{1}{u}$ by $\frac{1}{8}$ .

$3 \int - \frac{1}{u \cdot 8} d u = \int \frac{1}{x} d x$

Step 2.2.3.3

Move $8$ to the left of $u$ .

$3 \int - \frac{1}{8 u} d u = \int \frac{1}{x} d x$

Step 2.2.4

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

$3 (- \int \frac{1}{8 u} d u) = \int \frac{1}{x} d x$

Step 2.2.5

Multiply $- 1$ by $3$ .

$- 3 \int \frac{1}{8 u} d u = \int \frac{1}{x} d x$

Step 2.2.6

Since $\frac{1}{8}$ is constant with respect to $u$ , move $\frac{1}{8}$ out of the integral.

$- 3 (\frac{1}{8} \int \frac{1}{u} d u) = \int \frac{1}{x} d x$

Step 2.2.7

Simplify.

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Step 2.2.7.1

Combine $\frac{1}{8}$ and $- 3$ .

$\frac{- 3}{8} \int \frac{1}{u} d u = \int \frac{1}{x} d x$

Step 2.2.7.2

Move the negative in front of the fraction.

$- \frac{3}{8} \int \frac{1}{u} d u = \int \frac{1}{x} d x$

Step 2.2.8

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

$- \frac{3}{8} (\ln (| u |) + C_{1}) = \int \frac{1}{x} d x$

Step 2.2.9

Simplify.

$- \frac{3}{8} \ln (| u |) + C_{1} = \int \frac{1}{x} d x$

Step 2.2.10

Replace all occurrences of $u$ with $(1 + 2 v) (1 - 2 v)$ .

$- \frac{3}{8} \ln (| (1 + 2 v) (1 - 2 v) |) + C_{1} = \int \frac{1}{x} d x$

Step 2.3

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

$- \frac{3}{8} \ln (| (1 + 2 v) (1 - 2 v) |) + C_{1} = \ln (| x |) + C_{2}$

Step 2.4

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

$- \frac{3}{8} \ln (| (1 + 2 v) (1 - 2 v) |) = \ln (| x |) + C$

Step 3

Solve for $v$ .

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Step 3.1

Multiply both sides of the equation by $- \frac{8}{3}$ .

$- \frac{8}{3} (- \frac{3}{8} \ln (| (1 + 2 v) (1 - 2 v) |)) = - \frac{8}{3} (\ln (| x |) + C)$

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 $- \frac{8}{3} (- \frac{3}{8} \ln (| (1 + 2 v) (1 - 2 v) |))$ .

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Step 3.2.1.1.1

Expand $(1 + 2 v) (1 - 2 v)$ using the FOIL Method.

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Step 3.2.1.1.1.1

Apply the distributive property.

$- \frac{8}{3} (- \frac{3}{8} \ln (| 1 (1 - 2 v) + 2 v (1 - 2 v) |)) = - \frac{8}{3} (\ln (| x |) + C)$

Step 3.2.1.1.1.2

Apply the distributive property.

$- \frac{8}{3} (- \frac{3}{8} \ln (| 1 \cdot 1 + 1 (- 2 v) + 2 v (1 - 2 v) |)) = - \frac{8}{3} (\ln (| x |) + C)$

Step 3.2.1.1.1.3

Apply the distributive property.

$- \frac{8}{3} (- \frac{3}{8} \ln (| 1 \cdot 1 + 1 (- 2 v) + 2 v \cdot 1 + 2 v (- 2 v) |)) = - \frac{8}{3} (\ln (| x |) + C)$

Step 3.2.1.1.2

Simplify and combine like terms.

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Step 3.2.1.1.2.1

Simplify each term.

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Step 3.2.1.1.2.1.1

Multiply $1$ by $1$ .

$- \frac{8}{3} (- \frac{3}{8} \ln (| 1 + 1 (- 2 v) + 2 v \cdot 1 + 2 v (- 2 v) |)) = - \frac{8}{3} (\ln (| x |) + C)$

Step 3.2.1.1.2.1.2

Multiply $- 2 v$ by $1$ .

$- \frac{8}{3} (- \frac{3}{8} \ln (| 1 - 2 v + 2 v \cdot 1 + 2 v (- 2 v) |)) = - \frac{8}{3} (\ln (| x |) + C)$

Step 3.2.1.1.2.1.3

Multiply $2$ by $1$ .

$- \frac{8}{3} (- \frac{3}{8} \ln (| 1 - 2 v + 2 v + 2 v (- 2 v) |)) = - \frac{8}{3} (\ln (| x |) + C)$

Step 3.2.1.1.2.1.4

Rewrite using the commutative property of multiplication.

$- \frac{8}{3} (- \frac{3}{8} \ln (| 1 - 2 v + 2 v + 2 \cdot - 2 v \cdot v |)) = - \frac{8}{3} (\ln (| x |) + C)$

Step 3.2.1.1.2.1.5

Multiply $v$ by $v$ by adding the exponents.

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Step 3.2.1.1.2.1.5.1

Move $v$ .

$- \frac{8}{3} (- \frac{3}{8} \ln (| 1 - 2 v + 2 v + 2 \cdot - 2 (v \cdot v) |)) = - \frac{8}{3} (\ln (| x |) + C)$

Step 3.2.1.1.2.1.5.2

Multiply $v$ by $v$ .

$- \frac{8}{3} (- \frac{3}{8} \ln (| 1 - 2 v + 2 v + 2 \cdot - 2 v^{2} |)) = - \frac{8}{3} (\ln (| x |) + C)$

Step 3.2.1.1.2.1.6

Multiply $2$ by $- 2$ .

$- \frac{8}{3} (- \frac{3}{8} \ln (| 1 - 2 v + 2 v - 4 v^{2} |)) = - \frac{8}{3} (\ln (| x |) + C)$

Step 3.2.1.1.2.2

Add $- 2 v$ and $2 v$ .

$- \frac{8}{3} (- \frac{3}{8} \ln (| 1 + 0 - 4 v^{2} |)) = - \frac{8}{3} (\ln (| x |) + C)$

Step 3.2.1.1.2.3

Add $1$ and $0$ .

$- \frac{8}{3} (- \frac{3}{8} \ln (| 1 - 4 v^{2} |)) = - \frac{8}{3} (\ln (| x |) + C)$

Step 3.2.1.1.3

Combine $\ln (| 1 - 4 v^{2} |)$ and $\frac{3}{8}$ .

$- \frac{8}{3} (- \frac{\ln (| 1 - 4 v^{2} |) \cdot 3}{8}) = - \frac{8}{3} (\ln (| x |) + C)$

Step 3.2.1.1.4

Cancel the common factor of $8$ .

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Step 3.2.1.1.4.1

Move the leading negative in $- \frac{8}{3}$ into the numerator.

$\frac{- 8}{3} (- \frac{\ln (| 1 - 4 v^{2} |) \cdot 3}{8}) = - \frac{8}{3} (\ln (| x |) + C)$

Step 3.2.1.1.4.2

Move the leading negative in $- \frac{\ln (| 1 - 4 v^{2} |) \cdot 3}{8}$ into the numerator.

$\frac{- 8}{3} \cdot \frac{- \ln (| 1 - 4 v^{2} |) \cdot 3}{8} = - \frac{8}{3} (\ln (| x |) + C)$

Step 3.2.1.1.4.3

Factor $8$ out of $- 8$ .

$\frac{8 (- 1)}{3} \cdot \frac{- \ln (| 1 - 4 v^{2} |) \cdot 3}{8} = - \frac{8}{3} (\ln (| x |) + C)$

Step 3.2.1.1.4.4

Cancel the common factor.

$\frac{8 \cdot - 1}{3} \cdot \frac{- \ln (| 1 - 4 v^{2} |) \cdot 3}{8} = - \frac{8}{3} (\ln (| x |) + C)$

Step 3.2.1.1.4.5

Rewrite the expression.

$\frac{- 1}{3} (- \ln (| 1 - 4 v^{2} |) \cdot 3) = - \frac{8}{3} (\ln (| x |) + C)$

Step 3.2.1.1.5

Cancel the common factor of $3$ .

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Step 3.2.1.1.5.1

Factor $3$ out of $- \ln (| 1 - 4 v^{2} |) \cdot 3$ .

$\frac{- 1}{3} (3 \cdot (- \ln (| 1 - 4 v^{2} |))) = - \frac{8}{3} (\ln (| x |) + C)$

Step 3.2.1.1.5.2

Cancel the common factor.

$\frac{- 1}{3} (3 \cdot (- \ln (| 1 - 4 v^{2} |))) = - \frac{8}{3} (\ln (| x |) + C)$

Step 3.2.1.1.5.3

Rewrite the expression.

$- - \ln (| 1 - 4 v^{2} |) = - \frac{8}{3} (\ln (| x |) + C)$

Step 3.2.1.1.6

Multiply.

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Step 3.2.1.1.6.1

Multiply $- 1$ by $- 1$ .

$1 \ln (| 1 - 4 v^{2} |) = - \frac{8}{3} (\ln (| x |) + C)$

Step 3.2.1.1.6.2

Multiply $\ln (| 1 - 4 v^{2} |)$ by $1$ .

$\ln (| 1 - 4 v^{2} |) = - \frac{8}{3} (\ln (| x |) + C)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $- \frac{8}{3} (\ln (| x |) + C)$ .

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Step 3.2.2.1.1

Simplify terms.

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Step 3.2.2.1.1.1

Apply the distributive property.

$\ln (| 1 - 4 v^{2} |) = - \frac{8}{3} \ln (| x |) - \frac{8}{3} C$

Step 3.2.2.1.1.2

Combine $\ln (| x |)$ and $\frac{8}{3}$ .

$\ln (| 1 - 4 v^{2} |) = - \frac{\ln (| x |) \cdot 8}{3} - \frac{8}{3} C$

Step 3.2.2.1.1.3

Combine $C$ and $\frac{8}{3}$ .

$\ln (| 1 - 4 v^{2} |) = - \frac{\ln (| x |) \cdot 8}{3} - \frac{C \cdot 8}{3}$

Step 3.2.2.1.2

Simplify each term.

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Step 3.2.2.1.2.1

Move $8$ to the left of $\ln (| x |)$ .

$\ln (| 1 - 4 v^{2} |) = - \frac{8 \cdot \ln (| x |)}{3} - \frac{C \cdot 8}{3}$

Step 3.2.2.1.2.2

Move $8$ to the left of $C$ .

$\ln (| 1 - 4 v^{2} |) = - \frac{8 \ln (| x |)}{3} - \frac{8 C}{3}$

Step 3.3

Multiply each term in $\ln (| 1 - 4 v^{2} |) = - \frac{8 \ln (| x |)}{3} - \frac{8 C}{3}$ by $3$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $\ln (| 1 - 4 v^{2} |) = - \frac{8 \ln (| x |)}{3} - \frac{8 C}{3}$ by $3$ .

$\ln (| 1 - 4 v^{2} |) \cdot 3 = - \frac{8 \ln (| x |)}{3} \cdot 3 - \frac{8 C}{3} \cdot 3$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Move $3$ to the left of $\ln (| 1 - 4 v^{2} |)$ .

$3 \ln (| 1 - 4 v^{2} |) = - \frac{8 \ln (| x |)}{3} \cdot 3 - \frac{8 C}{3} \cdot 3$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

Cancel the common factor of $3$ .

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Step 3.3.3.1.1.1

Move the leading negative in $- \frac{8 \ln (| x |)}{3}$ into the numerator.

$3 \ln (| 1 - 4 v^{2} |) = \frac{- 8 \ln (| x |)}{3} \cdot 3 - \frac{8 C}{3} \cdot 3$

Step 3.3.3.1.1.2

Cancel the common factor.

$3 \ln (| 1 - 4 v^{2} |) = \frac{- 8 \ln (| x |)}{3} \cdot 3 - \frac{8 C}{3} \cdot 3$

Step 3.3.3.1.1.3

Rewrite the expression.

$3 \ln (| 1 - 4 v^{2} |) = - 8 \ln (| x |) - \frac{8 C}{3} \cdot 3$

Step 3.3.3.1.2

Cancel the common factor of $3$ .

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Step 3.3.3.1.2.1

Move the leading negative in $- \frac{8 C}{3}$ into the numerator.

$3 \ln (| 1 - 4 v^{2} |) = - 8 \ln (| x |) + \frac{- 8 C}{3} \cdot 3$

Step 3.3.3.1.2.2

Cancel the common factor.

$3 \ln (| 1 - 4 v^{2} |) = - 8 \ln (| x |) + \frac{- 8 C}{3} \cdot 3$

Step 3.3.3.1.2.3

Rewrite the expression.

$3 \ln (| 1 - 4 v^{2} |) = - 8 \ln (| x |) - 8 C$

Step 3.4

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

$3 \ln (| 1 - 4 v^{2} |) + 8 \ln (| x |) = - 8 C$

Step 3.5

Simplify the left side.

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Step 3.5.1

Simplify $3 \ln (| 1 - 4 v^{2} |) + 8 \ln (| x |)$ .

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Step 3.5.1.1

Simplify each term.

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Step 3.5.1.1.1

Simplify $3 \ln (| 1 - 4 v^{2} |)$ by moving $3$ inside the logarithm.

$\ln ({| 1 - 4 v^{2} |}^{3}) + 8 \ln (| x |) = - 8 C$

Step 3.5.1.1.2

Simplify $8 \ln (| x |)$ by moving $8$ inside the logarithm.

$\ln ({| 1 - 4 v^{2} |}^{3}) + \ln ({| x |}^{8}) = - 8 C$

Step 3.5.1.1.3

Remove the absolute value in ${| x |}^{8}$ because exponentiations with even powers are always positive.

$\ln ({| 1 - 4 v^{2} |}^{3}) + \ln (x^{8}) = - 8 C$

Step 3.5.1.2

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

$\ln ({| 1 - 4 v^{2} |}^{3} x^{8}) = - 8 C$

Step 3.5.1.3

Reorder factors in $\ln ({| 1 - 4 v^{2} |}^{3} x^{8})$ .

$\ln (x^{8} {| 1 - 4 v^{2} |}^{3}) = - 8 C$

Step 3.6

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

$e^{\ln (x^{8} {| 1 - 4 v^{2} |}^{3})} = e^{- 8 C}$

Step 3.7

Rewrite $\ln (x^{8} {| 1 - 4 v^{2} |}^{3}) = - 8 C$ 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 $b^{y} = x$ .

$e^{- 8 C} = x^{8} {| 1 - 4 v^{2} |}^{3}$

Step 3.8

Solve for $v$ .

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Step 3.8.1

Rewrite the equation as $x^{8} {| 1 - 4 v^{2} |}^{3} = e^{- 8 C}$ .

$x^{8} {| 1 - 4 v^{2} |}^{3} = e^{- 8 C}$

Step 3.8.2

Divide each term in $x^{8} {| 1 - 4 v^{2} |}^{3} = e^{- 8 C}$ by $x^{8}$ and simplify.

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Step 3.8.2.1

Divide each term in $x^{8} {| 1 - 4 v^{2} |}^{3} = e^{- 8 C}$ by $x^{8}$ .

$\frac{x^{8} {| 1 - 4 v^{2} |}^{3}}{x^{8}} = \frac{e^{- 8 C}}{x^{8}}$

Step 3.8.2.2

Simplify the left side.

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Step 3.8.2.2.1

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

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Step 3.8.2.2.1.1

Cancel the common factor.

$\frac{x^{8} {| 1 - 4 v^{2} |}^{3}}{x^{8}} = \frac{e^{- 8 C}}{x^{8}}$

Step 3.8.2.2.1.2

Divide ${| 1 - 4 v^{2} |}^{3}$ by $1$ .

${| 1 - 4 v^{2} |}^{3} = \frac{e^{- 8 C}}{x^{8}}$

Step 3.8.3

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

$| 1 - 4 v^{2} | = \sqrt[3]{\frac{e^{- 8 C}}{x^{8}}}$

Step 3.8.4

Simplify $\sqrt[3]{\frac{e^{- 8 C}}{x^{8}}}$ .

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Step 3.8.4.1

Rewrite $\frac{e^{- 8 C}}{x^{8}}$ as ${(\frac{1}{x^{2}})}^{3} \frac{e^{- 8 C}}{x^{2}}$ .

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Step 3.8.4.1.1

Factor the perfect power $1^{3}$ out of $e^{- 8 C}$ .

$| 1 - 4 v^{2} | = \sqrt[3]{\frac{1^{3} e^{- 8 C}}{x^{8}}}$

Step 3.8.4.1.2

Factor the perfect power ${(x^{2})}^{3}$ out of $x^{8}$ .

$| 1 - 4 v^{2} | = \sqrt[3]{\frac{1^{3} e^{- 8 C}}{{(x^{2})}^{3} x^{2}}}$

Step 3.8.4.1.3

Rearrange the fraction $\frac{1^{3} e^{- 8 C}}{{(x^{2})}^{3} x^{2}}$ .

$| 1 - 4 v^{2} | = \sqrt[3]{{(\frac{1}{x^{2}})}^{3} \frac{e^{- 8 C}}{x^{2}}}$

Step 3.8.4.2

Pull terms out from under the radical.

$| 1 - 4 v^{2} | = \frac{1}{x^{2}} \sqrt[3]{\frac{e^{- 8 C}}{x^{2}}}$

Step 3.8.4.3

Rewrite $\sqrt[3]{\frac{e^{- 8 C}}{x^{2}}}$ as $\frac{\sqrt[3]{e^{- 8 C}}}{\sqrt[3]{x^{2}}}$ .

$| 1 - 4 v^{2} | = \frac{1}{x^{2}} \cdot \frac{\sqrt[3]{e^{- 8 C}}}{\sqrt[3]{x^{2}}}$

Step 3.8.4.4

Combine.

$| 1 - 4 v^{2} | = \frac{1 \sqrt[3]{e^{- 8 C}}}{x^{2} \sqrt[3]{x^{2}}}$

Step 3.8.4.5

Multiply $\sqrt[3]{e^{- 8 C}}$ by $1$ .

$| 1 - 4 v^{2} | = \frac{\sqrt[3]{e^{- 8 C}}}{x^{2} \sqrt[3]{x^{2}}}$

Step 3.8.4.6

Multiply $\frac{\sqrt[3]{e^{- 8 C}}}{x^{2} \sqrt[3]{x^{2}}}$ by $\frac{{\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{2}}$ .

$| 1 - 4 v^{2} | = \frac{\sqrt[3]{e^{- 8 C}}}{x^{2} \sqrt[3]{x^{2}}} \cdot \frac{{\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{2}}$

Step 3.8.4.7

Combine and simplify the denominator.

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Step 3.8.4.7.1

Multiply $\frac{\sqrt[3]{e^{- 8 C}}}{x^{2} \sqrt[3]{x^{2}}}$ by $\frac{{\sqrt[3]{x^{2}}}^{2}}{{\sqrt[3]{x^{2}}}^{2}}$ .

$| 1 - 4 v^{2} | = \frac{\sqrt[3]{e^{- 8 C}} {\sqrt[3]{x^{2}}}^{2}}{x^{2} \sqrt[3]{x^{2}} {\sqrt[3]{x^{2}}}^{2}}$

Step 3.8.4.7.2

Move $\sqrt[3]{x^{2}}$ .

$| 1 - 4 v^{2} | = \frac{\sqrt[3]{e^{- 8 C}} {\sqrt[3]{x^{2}}}^{2}}{x^{2} (\sqrt[3]{x^{2}} {\sqrt[3]{x^{2}}}^{2})}$

Step 3.8.4.7.3

Raise $\sqrt[3]{x^{2}}$ to the power of $1$ .

$| 1 - 4 v^{2} | = \frac{\sqrt[3]{e^{- 8 C}} {\sqrt[3]{x^{2}}}^{2}}{x^{2} ({\sqrt[3]{x^{2}}}^{1} {\sqrt[3]{x^{2}}}^{2})}$

Step 3.8.4.7.4

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

$| 1 - 4 v^{2} | = \frac{\sqrt[3]{e^{- 8 C}} {\sqrt[3]{x^{2}}}^{2}}{x^{2} {\sqrt[3]{x^{2}}}^{1 + 2}}$

Step 3.8.4.7.5

Add $1$ and $2$ .

$| 1 - 4 v^{2} | = \frac{\sqrt[3]{e^{- 8 C}} {\sqrt[3]{x^{2}}}^{2}}{x^{2} {\sqrt[3]{x^{2}}}^{3}}$

Step 3.8.4.7.6

Rewrite ${\sqrt[3]{x^{2}}}^{3}$ as $x^{2}$ .

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Step 3.8.4.7.6.1

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

$| 1 - 4 v^{2} | = \frac{\sqrt[3]{e^{- 8 C}} {\sqrt[3]{x^{2}}}^{2}}{x^{2} {(x^{\frac{2}{3}})}^{3}}$

Step 3.8.4.7.6.2

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

$| 1 - 4 v^{2} | = \frac{\sqrt[3]{e^{- 8 C}} {\sqrt[3]{x^{2}}}^{2}}{x^{2} x^{\frac{2}{3} \cdot 3}}$

Step 3.8.4.7.6.3

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

$| 1 - 4 v^{2} | = \frac{\sqrt[3]{e^{- 8 C}} {\sqrt[3]{x^{2}}}^{2}}{x^{2} x^{\frac{2 \cdot 3}{3}}}$

Step 3.8.4.7.6.4

Multiply $2$ by $3$ .

$| 1 - 4 v^{2} | = \frac{\sqrt[3]{e^{- 8 C}} {\sqrt[3]{x^{2}}}^{2}}{x^{2} x^{\frac{6}{3}}}$

Step 3.8.4.7.6.5

Cancel the common factor of $6$ and $3$ .

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Step 3.8.4.7.6.5.1

Factor $3$ out of $6$ .

$| 1 - 4 v^{2} | = \frac{\sqrt[3]{e^{- 8 C}} {\sqrt[3]{x^{2}}}^{2}}{x^{2} x^{\frac{3 \cdot 2}{3}}}$

Step 3.8.4.7.6.5.2

Cancel the common factors.

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Step 3.8.4.7.6.5.2.1

Factor $3$ out of $3$ .

$| 1 - 4 v^{2} | = \frac{\sqrt[3]{e^{- 8 C}} {\sqrt[3]{x^{2}}}^{2}}{x^{2} x^{\frac{3 \cdot 2}{3 (1)}}}$

Step 3.8.4.7.6.5.2.2

Cancel the common factor.

$| 1 - 4 v^{2} | = \frac{\sqrt[3]{e^{- 8 C}} {\sqrt[3]{x^{2}}}^{2}}{x^{2} x^{\frac{3 \cdot 2}{3 \cdot 1}}}$

Step 3.8.4.7.6.5.2.3

Rewrite the expression.

$| 1 - 4 v^{2} | = \frac{\sqrt[3]{e^{- 8 C}} {\sqrt[3]{x^{2}}}^{2}}{x^{2} x^{\frac{2}{1}}}$

Step 3.8.4.7.6.5.2.4

Divide $2$ by $1$ .

$| 1 - 4 v^{2} | = \frac{\sqrt[3]{e^{- 8 C}} {\sqrt[3]{x^{2}}}^{2}}{x^{2} x^{2}}$

Step 3.8.4.8

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

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Step 3.8.4.8.1

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

$| 1 - 4 v^{2} | = \frac{\sqrt[3]{e^{- 8 C}} {\sqrt[3]{x^{2}}}^{2}}{x^{2 + 2}}$

Step 3.8.4.8.2

Add $2$ and $2$ .

$| 1 - 4 v^{2} | = \frac{\sqrt[3]{e^{- 8 C}} {\sqrt[3]{x^{2}}}^{2}}{x^{4}}$

Step 3.8.4.9

Simplify the numerator.

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Step 3.8.4.9.1

Rewrite ${\sqrt[3]{x^{2}}}^{2}$ as $\sqrt[3]{{(x^{2})}^{2}}$ .

$| 1 - 4 v^{2} | = \frac{\sqrt[3]{e^{- 8 C}} \sqrt[3]{{(x^{2})}^{2}}}{x^{4}}$

Step 3.8.4.9.2

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

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Step 3.8.4.9.2.1

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

$| 1 - 4 v^{2} | = \frac{\sqrt[3]{e^{- 8 C}} \sqrt[3]{x^{2 \cdot 2}}}{x^{4}}$

Step 3.8.4.9.2.2

Multiply $2$ by $2$ .

$| 1 - 4 v^{2} | = \frac{\sqrt[3]{e^{- 8 C}} \sqrt[3]{x^{4}}}{x^{4}}$

Step 3.8.4.9.3

Factor out $x^{3}$ .

$| 1 - 4 v^{2} | = \frac{\sqrt[3]{e^{- 8 C}} \sqrt[3]{x^{3} x}}{x^{4}}$

Step 3.8.4.9.4

Pull terms out from under the radical.

$| 1 - 4 v^{2} | = \frac{\sqrt[3]{e^{- 8 C}} x \sqrt[3]{x}}{x^{4}}$

Step 3.8.4.9.5

Combine using the product rule for radicals.

$| 1 - 4 v^{2} | = \frac{x \sqrt[3]{e^{- 8 C} x}}{x^{4}}$

Step 3.8.4.10

Reduce the expression by cancelling the common factors.

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Step 3.8.4.10.1

Cancel the common factor of $x$ and $x^{4}$ .

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Step 3.8.4.10.1.1

Factor $x$ out of $x \sqrt[3]{e^{- 8 C} x}$ .

$| 1 - 4 v^{2} | = \frac{x \sqrt[3]{e^{- 8 C} x}}{x^{4}}$

Step 3.8.4.10.1.2

Cancel the common factors.

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Step 3.8.4.10.1.2.1

Factor $x$ out of $x^{4}$ .

$| 1 - 4 v^{2} | = \frac{x \sqrt[3]{e^{- 8 C} x}}{x \cdot x^{3}}$

Step 3.8.4.10.1.2.2

Cancel the common factor.

$| 1 - 4 v^{2} | = \frac{x \sqrt[3]{e^{- 8 C} x}}{x \cdot x^{3}}$

Step 3.8.4.10.1.2.3

Rewrite the expression.

$| 1 - 4 v^{2} | = \frac{\sqrt[3]{e^{- 8 C} x}}{x^{3}}$

Step 3.8.4.10.2

Reorder factors in $\frac{\sqrt[3]{e^{- 8 C} x}}{x^{3}}$ .

$| 1 - 4 v^{2} | = \frac{\sqrt[3]{x e^{- 8 C}}}{x^{3}}$

Step 3.8.5

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

$1 - 4 v^{2} = \pm \frac{\sqrt[3]{x e^{- 8 C}}}{x^{3}}$

Step 3.8.6

Subtract $1$ from both sides of the equation.

$- 4 v^{2} = \pm \frac{\sqrt[3]{x e^{- 8 C}}}{x^{3}} - 1$

Step 3.8.7

Divide each term in $- 4 v^{2} = \pm \frac{\sqrt[3]{x e^{- 8 C}}}{x^{3}} - 1$ by $- 4$ and simplify.

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Step 3.8.7.1

Divide each term in $- 4 v^{2} = \pm \frac{\sqrt[3]{x e^{- 8 C}}}{x^{3}} - 1$ by $- 4$ .

$\frac{- 4 v^{2}}{- 4} = \frac{\pm \frac{\sqrt[3]{x e^{- 8 C}}}{x^{3}}}{- 4} + \frac{- 1}{- 4}$

Step 3.8.7.2

Simplify the left side.

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Step 3.8.7.2.1

Cancel the common factor of $- 4$ .

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Step 3.8.7.2.1.1

Cancel the common factor.

$\frac{- 4 v^{2}}{- 4} = \frac{\pm \frac{\sqrt[3]{x e^{- 8 C}}}{x^{3}}}{- 4} + \frac{- 1}{- 4}$

Step 3.8.7.2.1.2

Divide $v^{2}$ by $1$ .

$v^{2} = \frac{\pm \frac{\sqrt[3]{x e^{- 8 C}}}{x^{3}}}{- 4} + \frac{- 1}{- 4}$

Step 3.8.7.3

Simplify the right side.

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Step 3.8.7.3.1

Simplify each term.

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Step 3.8.7.3.1.1

Simplify $\frac{\pm \frac{\sqrt[3]{x e^{- 8 C}}}{x^{3}}}{- 4}$ .

$v^{2} = \frac{\pm \frac{\sqrt[3]{x e^{- 8 C}}}{x^{3}}}{4} + \frac{- 1}{- 4}$

Step 3.8.7.3.1.2

Dividing two negative values results in a positive value.

$v^{2} = \frac{\pm \frac{\sqrt[3]{x e^{- 8 C}}}{x^{3}}}{4} + \frac{1}{4}$

Step 3.8.8

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

$v = \pm \sqrt{\frac{\pm \frac{\sqrt[3]{x e^{- 8 C}}}{x^{3}}}{4} + \frac{1}{4}}$

Step 3.8.9

Simplify $\pm \sqrt{\frac{\pm \frac{\sqrt[3]{x e^{- 8 C}}}{x^{3}}}{4} + \frac{1}{4}}$ .

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Step 3.8.9.1

Combine the numerators over the common denominator.

$v = \pm \sqrt{\frac{\pm \frac{\sqrt[3]{x e^{- 8 C}}}{x^{3}} + 1}{4}}$

Step 3.8.9.2

Rewrite $\frac{\pm \frac{\sqrt[3]{x e^{- 8 C}}}{x^{3}} + 1}{4}$ as ${(\frac{1}{2})}^{2} (\pm \frac{\sqrt[3]{x e^{- 8 C}}}{x^{3}} + 1)$ .

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Step 3.8.9.2.1

Factor the perfect power $1^{2}$ out of $\pm \frac{\sqrt[3]{x e^{- 8 C}}}{x^{3}} + 1$ .

$v = \pm \sqrt{\frac{1^{2} (\pm \frac{\sqrt[3]{x e^{- 8 C}}}{x^{3}} + 1)}{4}}$

Step 3.8.9.2.2

Factor the perfect power $2^{2}$ out of $4$ .

$v = \pm \sqrt{\frac{1^{2} (\pm \frac{\sqrt[3]{x e^{- 8 C}}}{x^{3}} + 1)}{2^{2} \cdot 1}}$

Step 3.8.9.2.3

Rearrange the fraction $\frac{1^{2} (\pm \frac{\sqrt[3]{x e^{- 8 C}}}{x^{3}} + 1)}{2^{2} \cdot 1}$ .

$v = \pm \sqrt{{(\frac{1}{2})}^{2} (\pm \frac{\sqrt[3]{x e^{- 8 C}}}{x^{3}} + 1)}$

Step 3.8.9.3

Pull terms out from under the radical.

$v = \pm \frac{1}{2} \sqrt{\pm \frac{\sqrt[3]{x e^{- 8 C}}}{x^{3}} + 1}$

Step 3.8.9.4

Combine $\frac{1}{2}$ and $\sqrt{\pm \frac{\sqrt[3]{x e^{- 8 C}}}{x^{3}} + 1}$ .

$v = \pm \frac{\sqrt{\pm \frac{\sqrt[3]{x e^{- 8 C}}}{x^{3}} + 1}}{2}$

Step 4

Simplify the constant of integration.

$v = \pm \frac{\sqrt{\pm \frac{\sqrt[3]{x C}}{x^{3}} + 1}}{2}$