Calculus Examples

$4 x v d v + (3 v^{2} - 1) d x = 0$

Step 1

Subtract $(3 v^{2} - 1) d x$ from both sides of the equation.

$4 x v d v = - (3 v^{2} - 1) d x$

Step 2

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

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

Step 3

Simplify.

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Step 3.1

Rewrite using the commutative property of multiplication.

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

Step 3.2

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

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

Step 3.3

Cancel the common factor of $x$ .

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Step 3.3.1

Factor $x$ out of $x v$ .

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 3.4

Combine $\frac{4}{3 v^{2} - 1}$ and $v$ .

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

Step 3.5

Rewrite using the commutative property of multiplication.

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

Step 3.6

Cancel the common factor of $3 v^{2} - 1$ .

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Step 3.6.1

Move the leading negative in $- \frac{1}{x (3 v^{2} - 1)}$ into the numerator.

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

Step 3.6.2

Factor $3 v^{2} - 1$ out of $x (3 v^{2} - 1)$ .

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

Step 3.6.3

Cancel the common factor.

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

Step 3.6.4

Rewrite the expression.

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

Step 3.7

Move the negative in front of the fraction.

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

Integrate the left side.

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Step 4.2.1

Since $4$ is constant with respect to $v$ , move $4$ out of the integral.

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

Step 4.2.2

Let $u = 3 v^{2} - 1$ . Then $d u = 6 v d v$ , so $\frac{1}{6} d u = v d v$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.2.2.1

Let $u = 3 v^{2} - 1$ . Find $\frac{d u}{d v}$ .

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Step 4.2.2.1.1

Differentiate $3 v^{2} - 1$ .

$\frac{d}{d v} [3 v^{2} - 1]$

Step 4.2.2.1.2

By the Sum Rule, the derivative of $3 v^{2} - 1$ with respect to $v$ is $\frac{d}{d v} [3 v^{2}] + \frac{d}{d v} [- 1]$ .

$\frac{d}{d v} [3 v^{2}] + \frac{d}{d v} [- 1]$

Step 4.2.2.1.3

Evaluate $\frac{d}{d v} [3 v^{2}]$ .

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Step 4.2.2.1.3.1

Since $3$ is constant with respect to $v$ , the derivative of $3 v^{2}$ with respect to $v$ is $3 \frac{d}{d v} [v^{2}]$ .

$3 \frac{d}{d v} [v^{2}] + \frac{d}{d v} [- 1]$

Step 4.2.2.1.3.2

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

$3 (2 v) + \frac{d}{d v} [- 1]$

Step 4.2.2.1.3.3

Multiply $2$ by $3$ .

$6 v + \frac{d}{d v} [- 1]$

Step 4.2.2.1.4

Differentiate using the Constant Rule.

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Step 4.2.2.1.4.1

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

$6 v + 0$

Step 4.2.2.1.4.2

Add $6 v$ and $0$ .

$6 v$

Step 4.2.2.2

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

$4 \int \frac{1}{u} \cdot \frac{1}{6} d u = \int - \frac{1}{x} d x$

Step 4.2.3

Simplify.

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Step 4.2.3.1

Multiply $\frac{1}{u}$ by $\frac{1}{6}$ .

$4 \int \frac{1}{u \cdot 6} d u = \int - \frac{1}{x} d x$

Step 4.2.3.2

Move $6$ to the left of $u$ .

$4 \int \frac{1}{6 u} d u = \int - \frac{1}{x} d x$

Step 4.2.4

Since $\frac{1}{6}$ is constant with respect to $u$ , move $\frac{1}{6}$ out of the integral.

$4 (\frac{1}{6} \int \frac{1}{u} d u) = \int - \frac{1}{x} d x$

Step 4.2.5

Simplify.

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Step 4.2.5.1

Combine $\frac{1}{6}$ and $4$ .

$\frac{4}{6} \int \frac{1}{u} d u = \int - \frac{1}{x} d x$

Step 4.2.5.2

Cancel the common factor of $4$ and $6$ .

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Step 4.2.5.2.1

Factor $2$ out of $4$ .

$\frac{2 (2)}{6} \int \frac{1}{u} d u = \int - \frac{1}{x} d x$

Step 4.2.5.2.2

Cancel the common factors.

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Step 4.2.5.2.2.1

Factor $2$ out of $6$ .

$\frac{2 \cdot 2}{2 \cdot 3} \int \frac{1}{u} d u = \int - \frac{1}{x} d x$

Step 4.2.5.2.2.2

Cancel the common factor.

$\frac{2 \cdot 2}{2 \cdot 3} \int \frac{1}{u} d u = \int - \frac{1}{x} d x$

Step 4.2.5.2.2.3

Rewrite the expression.

$\frac{2}{3} \int \frac{1}{u} d u = \int - \frac{1}{x} d x$

Step 4.2.6

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

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

Step 4.2.7

Simplify.

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

Step 4.2.8

Replace all occurrences of $u$ with $3 v^{2} - 1$ .

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

Step 4.3

Integrate the right side.

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

Simplify.

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

Step 4.4

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

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

Step 5

Solve for $v$ .

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Step 5.1

Multiply both sides of the equation by $\frac{3}{2}$ .

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

Step 5.2

Simplify both sides of the equation.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

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

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Step 5.2.1.1.1

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

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

Step 5.2.1.1.2

Combine.

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

Step 5.2.1.1.3

Cancel the common factor of $3$ .

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Step 5.2.1.1.3.1

Cancel the common factor.

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

Step 5.2.1.1.3.2

Rewrite the expression.

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

Step 5.2.1.1.4

Cancel the common factor of $2$ .

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Step 5.2.1.1.4.1

Cancel the common factor.

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

Step 5.2.1.1.4.2

Divide $\ln (| 3 v^{2} - 1 |)$ by $1$ .

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

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

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

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Step 5.2.2.1.1

Apply the distributive property.

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

Step 5.2.2.1.2

Combine $\frac{3}{2}$ and $\ln (| x |)$ .

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

Step 5.2.2.1.3

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

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

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

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

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Simplify each term.

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Step 5.3.3.1.1

Cancel the common factor of $2$ .

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Step 5.3.3.1.1.1

Move the leading negative in $- \frac{3 \ln (| x |)}{2}$ into the numerator.

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

Step 5.3.3.1.1.2

Cancel the common factor.

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

Step 5.3.3.1.1.3

Rewrite the expression.

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

Step 5.3.3.1.2

Cancel the common factor of $2$ .

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Step 5.3.3.1.2.1

Cancel the common factor.

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

Step 5.3.3.1.2.2

Rewrite the expression.

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

Step 5.4

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

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

Step 5.5

Simplify the left side.

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Step 5.5.1

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

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Step 5.5.1.1

Simplify each term.

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Step 5.5.1.1.1

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

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

Step 5.5.1.1.2

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

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

Step 5.5.1.1.3

Simplify $3 \ln (| x |)$ by moving $3$ inside the logarithm.

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

Step 5.5.1.2

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

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

Step 5.5.1.3

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

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

Step 5.6

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

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

Step 5.7

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

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

Step 5.8

Solve for $v$ .

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Step 5.8.1

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

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

Step 5.8.2

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

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Step 5.8.2.1

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

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

Step 5.8.2.2

Simplify the left side.

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Step 5.8.2.2.1

Cancel the common factor of ${| x |}^{3}$ .

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Step 5.8.2.2.1.1

Cancel the common factor.

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

Step 5.8.2.2.1.2

Divide ${(3 v^{2} - 1)}^{2}$ by $1$ .

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

Step 5.8.3

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

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

Step 5.8.4

Simplify $\pm \sqrt{\frac{e^{3 C}}{{| x |}^{3}}}$ .

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Step 5.8.4.1

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

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Step 5.8.4.1.1

Factor the perfect power $1^{2}$ out of $e^{3 C}$ .

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

Step 5.8.4.1.2

Factor the perfect power ${| x |}^{2}$ out of ${| x |}^{3}$ .

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

Step 5.8.4.1.3

Rearrange the fraction $\frac{1^{2} e^{3 C}}{{| x |}^{2} | x |}$ .

$3 v^{2} - 1 = \pm \sqrt{{(\frac{1}{| x |})}^{2} \frac{e^{3 C}}{| x |}}$

Step 5.8.4.2

Pull terms out from under the radical.

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

Step 5.8.4.3

Rewrite $\sqrt{\frac{e^{3 C}}{| x |}}$ as $\frac{\sqrt{e^{3 C}}}{\sqrt{| x |}}$ .

$3 v^{2} - 1 = \pm \frac{1}{| x |} \cdot \frac{\sqrt{e^{3 C}}}{\sqrt{| x |}}$

Step 5.8.4.4

Combine.

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

Step 5.8.4.5

Multiply $\sqrt{e^{3 C}}$ by $1$ .

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

Step 5.8.4.6

Multiply $\frac{\sqrt{e^{3 C}}}{| x | \sqrt{| x |}}$ by $\frac{\sqrt{| x |}}{\sqrt{| x |}}$ .

$3 v^{2} - 1 = \pm \frac{\sqrt{e^{3 C}}}{| x | \sqrt{| x |}} \cdot \frac{\sqrt{| x |}}{\sqrt{| x |}}$

Step 5.8.4.7

Combine and simplify the denominator.

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Step 5.8.4.7.1

Multiply $\frac{\sqrt{e^{3 C}}}{| x | \sqrt{| x |}}$ by $\frac{\sqrt{| x |}}{\sqrt{| x |}}$ .

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

Step 5.8.4.7.2

Move $\sqrt{| x |}$ .

$3 v^{2} - 1 = \pm \frac{\sqrt{e^{3 C}} \sqrt{| x |}}{| x | (\sqrt{| x |} \sqrt{| x |})}$

Step 5.8.4.7.3

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

$3 v^{2} - 1 = \pm \frac{\sqrt{e^{3 C}} \sqrt{| x |}}{| x | ({\sqrt{| x |}}^{1} \sqrt{| x |})}$

Step 5.8.4.7.4

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

$3 v^{2} - 1 = \pm \frac{\sqrt{e^{3 C}} \sqrt{| x |}}{| x | ({\sqrt{| x |}}^{1} {\sqrt{| x |}}^{1})}$

Step 5.8.4.7.5

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

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

Step 5.8.4.7.6

Add $1$ and $1$ .

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

Step 5.8.4.7.7

Rewrite ${\sqrt{| x |}}^{2}$ as $| x |$ .

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Step 5.8.4.7.7.1

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

$3 v^{2} - 1 = \pm \frac{\sqrt{e^{3 C}} \sqrt{| x |}}{| x | {({| x |}^{\frac{1}{2}})}^{2}}$

Step 5.8.4.7.7.2

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

$3 v^{2} - 1 = \pm \frac{\sqrt{e^{3 C}} \sqrt{| x |}}{| x | {| x |}^{\frac{1}{2} \cdot 2}}$

Step 5.8.4.7.7.3

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

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

Step 5.8.4.7.7.4

Cancel the common factor of $2$ .

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Step 5.8.4.7.7.4.1

Cancel the common factor.

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

Step 5.8.4.7.7.4.2

Rewrite the expression.

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

Step 5.8.4.7.7.5

Simplify.

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

Step 5.8.4.8

Combine using the product rule for radicals.

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

Step 5.8.4.9

To multiply absolute values, multiply the terms inside each absolute value.

$3 v^{2} - 1 = \pm \frac{\sqrt{e^{3 C} | x |}}{| x \cdot x |}$

Step 5.8.4.10

Simplify the denominator.

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Step 5.8.4.10.1

Multiply $x$ by $x$ .

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

Step 5.8.4.10.2

Remove non-negative terms from the absolute value.

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

Step 5.8.5

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

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Step 5.8.5.1

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

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

Step 5.8.5.2

Add $1$ to both sides of the equation.

$3 v^{2} = \frac{\sqrt{e^{3 C} | x |}}{x^{2}} + 1$

Step 5.8.5.3

Divide each term in $3 v^{2} = \frac{\sqrt{e^{3 C} | x |}}{x^{2}} + 1$ by $3$ and simplify.

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Step 5.8.5.3.1

Divide each term in $3 v^{2} = \frac{\sqrt{e^{3 C} | x |}}{x^{2}} + 1$ by $3$ .

$\frac{3 v^{2}}{3} = \frac{\frac{\sqrt{e^{3 C} | x |}}{x^{2}}}{3} + \frac{1}{3}$

Step 5.8.5.3.2

Simplify the left side.

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Step 5.8.5.3.2.1

Cancel the common factor of $3$ .

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Step 5.8.5.3.2.1.1

Cancel the common factor.

$\frac{3 v^{2}}{3} = \frac{\frac{\sqrt{e^{3 C} | x |}}{x^{2}}}{3} + \frac{1}{3}$

Step 5.8.5.3.2.1.2

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

$v^{2} = \frac{\frac{\sqrt{e^{3 C} | x |}}{x^{2}}}{3} + \frac{1}{3}$

Step 5.8.5.3.3

Simplify the right side.

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Step 5.8.5.3.3.1

Simplify each term.

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Step 5.8.5.3.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$v^{2} = \frac{\sqrt{e^{3 C} | x |}}{x^{2}} \cdot \frac{1}{3} + \frac{1}{3}$

Step 5.8.5.3.3.1.2

Combine.

$v^{2} = \frac{\sqrt{e^{3 C} | x |} \cdot 1}{x^{2} \cdot 3} + \frac{1}{3}$

Step 5.8.5.3.3.1.3

Multiply $\sqrt{e^{3 C} | x |}$ by $1$ .

$v^{2} = \frac{\sqrt{e^{3 C} | x |}}{x^{2} \cdot 3} + \frac{1}{3}$

Step 5.8.5.3.3.1.4

Move $3$ to the left of $x^{2}$ .

$v^{2} = \frac{\sqrt{e^{3 C} | x |}}{3 x^{2}} + \frac{1}{3}$

Step 5.8.5.4

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

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

Step 5.8.5.5

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

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Step 5.8.5.5.1

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$v = \pm \sqrt{\frac{\sqrt{e^{3 C} | x |}}{3 x^{2}} + \frac{1}{3} \cdot \frac{x^{2}}{x^{2}}}$

Step 5.8.5.5.2

Multiply $\frac{1}{3}$ by $\frac{x^{2}}{x^{2}}$ .

$v = \pm \sqrt{\frac{\sqrt{e^{3 C} | x |}}{3 x^{2}} + \frac{x^{2}}{3 x^{2}}}$

Step 5.8.5.5.3

Combine the numerators over the common denominator.

$v = \pm \sqrt{\frac{\sqrt{e^{3 C} | x |} + x^{2}}{3 x^{2}}}$

Step 5.8.5.5.4

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

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Step 5.8.5.5.4.1

Factor the perfect power $1^{2}$ out of $\sqrt{e^{3 C} | x |} + x^{2}$ .

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

Step 5.8.5.5.4.2

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

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

Step 5.8.5.5.4.3

Rearrange the fraction $\frac{1^{2} (\sqrt{e^{3 C} | x |} + x^{2})}{x^{2} \cdot 3}$ .

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

Step 5.8.5.5.5

Pull terms out from under the radical.

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

Step 5.8.5.5.6

Rewrite $\sqrt{\frac{\sqrt{e^{3 C} | x |} + x^{2}}{3}}$ as $\frac{\sqrt{\sqrt{e^{3 C} | x |} + x^{2}}}{\sqrt{3}}$ .

$v = \pm \frac{1}{x} \cdot \frac{\sqrt{\sqrt{e^{3 C} | x |} + x^{2}}}{\sqrt{3}}$

Step 5.8.5.5.7

Combine.

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

Step 5.8.5.5.8

Multiply $\sqrt{\sqrt{e^{3 C} | x |} + x^{2}}$ by $1$ .

$v = \pm \frac{\sqrt{\sqrt{e^{3 C} | x |} + x^{2}}}{x \sqrt{3}}$

Step 5.8.5.5.9

Multiply $\frac{\sqrt{\sqrt{e^{3 C} | x |} + x^{2}}}{x \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$v = \pm \frac{\sqrt{\sqrt{e^{3 C} | x |} + x^{2}}}{x \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 5.8.5.5.10

Combine and simplify the denominator.

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Step 5.8.5.5.10.1

Multiply $\frac{\sqrt{\sqrt{e^{3 C} | x |} + x^{2}}}{x \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$v = \pm \frac{\sqrt{\sqrt{e^{3 C} | x |} + x^{2}} \sqrt{3}}{x \sqrt{3} \sqrt{3}}$

Step 5.8.5.5.10.2

Move $\sqrt{3}$ .

$v = \pm \frac{\sqrt{\sqrt{e^{3 C} | x |} + x^{2}} \sqrt{3}}{x (\sqrt{3} \sqrt{3})}$

Step 5.8.5.5.10.3

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

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

Step 5.8.5.5.10.4

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

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

Step 5.8.5.5.10.5

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

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

Step 5.8.5.5.10.6

Add $1$ and $1$ .

$v = \pm \frac{\sqrt{\sqrt{e^{3 C} | x |} + x^{2}} \sqrt{3}}{x {\sqrt{3}}^{2}}$

Step 5.8.5.5.10.7

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

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Step 5.8.5.5.10.7.1

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

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

Step 5.8.5.5.10.7.2

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

$v = \pm \frac{\sqrt{\sqrt{e^{3 C} | x |} + x^{2}} \sqrt{3}}{x \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 5.8.5.5.10.7.3

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

$v = \pm \frac{\sqrt{\sqrt{e^{3 C} | x |} + x^{2}} \sqrt{3}}{x \cdot 3^{\frac{2}{2}}}$

Step 5.8.5.5.10.7.4

Cancel the common factor of $2$ .

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Step 5.8.5.5.10.7.4.1

Cancel the common factor.

$v = \pm \frac{\sqrt{\sqrt{e^{3 C} | x |} + x^{2}} \sqrt{3}}{x \cdot 3^{\frac{2}{2}}}$

Step 5.8.5.5.10.7.4.2

Rewrite the expression.

$v = \pm \frac{\sqrt{\sqrt{e^{3 C} | x |} + x^{2}} \sqrt{3}}{x \cdot 3^{1}}$

Step 5.8.5.5.10.7.5

Evaluate the exponent.

$v = \pm \frac{\sqrt{\sqrt{e^{3 C} | x |} + x^{2}} \sqrt{3}}{x \cdot 3}$

Step 5.8.5.5.11

Combine using the product rule for radicals.

$v = \pm \frac{\sqrt{(\sqrt{e^{3 C} | x |} + x^{2}) \cdot 3}}{x \cdot 3}$

Step 5.8.5.5.12

Simplify the expression.

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Step 5.8.5.5.12.1

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

$v = \pm \frac{\sqrt{(\sqrt{e^{3 C} | x |} + x^{2}) \cdot 3}}{3 \cdot x}$

Step 5.8.5.5.12.2

Reorder factors in $\pm \frac{\sqrt{(\sqrt{e^{3 C} | x |} + x^{2}) \cdot 3}}{3 x}$ .

$v = \pm \frac{\sqrt{3 (\sqrt{e^{3 C} | x |} + x^{2})}}{3 x}$

Step 5.8.5.6

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

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Step 5.8.5.6.1

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

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

Step 5.8.5.6.2

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

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

Step 5.8.5.6.3

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

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

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

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

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

Step 5.8.5.7

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

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

Step 5.8.5.8

Add $1$ to both sides of the equation.

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

Step 5.8.5.9

Divide each term in $3 v^{2} = - \frac{\sqrt{e^{3 C} | x |}}{x^{2}} + 1$ by $3$ and simplify.

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Step 5.8.5.9.1

Divide each term in $3 v^{2} = - \frac{\sqrt{e^{3 C} | x |}}{x^{2}} + 1$ by $3$ .

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

Step 5.8.5.9.2

Simplify the left side.

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Step 5.8.5.9.2.1

Cancel the common factor of $3$ .

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Step 5.8.5.9.2.1.1

Cancel the common factor.

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

Step 5.8.5.9.2.1.2

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

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

Step 5.8.5.9.3

Simplify the right side.

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Step 5.8.5.9.3.1

Simplify each term.

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Step 5.8.5.9.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$v^{2} = - \frac{\sqrt{e^{3 C} | x |}}{x^{2}} \cdot \frac{1}{3} + \frac{1}{3}$

Step 5.8.5.9.3.1.2

Multiply $\frac{1}{3}$ by $\frac{\sqrt{e^{3 C} | x |}}{x^{2}}$ .

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

Step 5.8.5.10

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

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

Step 5.8.5.11

Simplify $\pm \sqrt{- \frac{\sqrt{e^{3 C} | x |}}{3 x^{2}} + \frac{1}{3}}$ .

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Step 5.8.5.11.1

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{x^{2}}{x^{2}}$ .

$v = \pm \sqrt{- \frac{\sqrt{e^{3 C} | x |}}{3 x^{2}} + \frac{1}{3} \cdot \frac{x^{2}}{x^{2}}}$

Step 5.8.5.11.2

Multiply $\frac{1}{3}$ by $\frac{x^{2}}{x^{2}}$ .

$v = \pm \sqrt{- \frac{\sqrt{e^{3 C} | x |}}{3 x^{2}} + \frac{x^{2}}{3 x^{2}}}$

Step 5.8.5.11.3

Combine the numerators over the common denominator.

$v = \pm \sqrt{\frac{- \sqrt{e^{3 C} | x |} + x^{2}}{3 x^{2}}}$

Step 5.8.5.11.4

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

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Step 5.8.5.11.4.1

Factor the perfect power $1^{2}$ out of $- \sqrt{e^{3 C} | x |} + x^{2}$ .

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

Step 5.8.5.11.4.2

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

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

Step 5.8.5.11.4.3

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

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

Step 5.8.5.11.5

Pull terms out from under the radical.

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

Step 5.8.5.11.6

Rewrite $\sqrt{\frac{- \sqrt{e^{3 C} | x |} + x^{2}}{3}}$ as $\frac{\sqrt{- \sqrt{e^{3 C} | x |} + x^{2}}}{\sqrt{3}}$ .

$v = \pm \frac{1}{x} \cdot \frac{\sqrt{- \sqrt{e^{3 C} | x |} + x^{2}}}{\sqrt{3}}$

Step 5.8.5.11.7

Combine.

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

Step 5.8.5.11.8

Multiply $\sqrt{- \sqrt{e^{3 C} | x |} + x^{2}}$ by $1$ .

$v = \pm \frac{\sqrt{- \sqrt{e^{3 C} | x |} + x^{2}}}{x \sqrt{3}}$

Step 5.8.5.11.9

Multiply $\frac{\sqrt{- \sqrt{e^{3 C} | x |} + x^{2}}}{x \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$v = \pm \frac{\sqrt{- \sqrt{e^{3 C} | x |} + x^{2}}}{x \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 5.8.5.11.10

Combine and simplify the denominator.

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Step 5.8.5.11.10.1

Multiply $\frac{\sqrt{- \sqrt{e^{3 C} | x |} + x^{2}}}{x \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$v = \pm \frac{\sqrt{- \sqrt{e^{3 C} | x |} + x^{2}} \sqrt{3}}{x \sqrt{3} \sqrt{3}}$

Step 5.8.5.11.10.2

Move $\sqrt{3}$ .

$v = \pm \frac{\sqrt{- \sqrt{e^{3 C} | x |} + x^{2}} \sqrt{3}}{x (\sqrt{3} \sqrt{3})}$

Step 5.8.5.11.10.3

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

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

Step 5.8.5.11.10.4

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

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

Step 5.8.5.11.10.5

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

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

Step 5.8.5.11.10.6

Add $1$ and $1$ .

$v = \pm \frac{\sqrt{- \sqrt{e^{3 C} | x |} + x^{2}} \sqrt{3}}{x {\sqrt{3}}^{2}}$

Step 5.8.5.11.10.7

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

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Step 5.8.5.11.10.7.1

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

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

Step 5.8.5.11.10.7.2

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

$v = \pm \frac{\sqrt{- \sqrt{e^{3 C} | x |} + x^{2}} \sqrt{3}}{x \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 5.8.5.11.10.7.3

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

$v = \pm \frac{\sqrt{- \sqrt{e^{3 C} | x |} + x^{2}} \sqrt{3}}{x \cdot 3^{\frac{2}{2}}}$

Step 5.8.5.11.10.7.4

Cancel the common factor of $2$ .

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Step 5.8.5.11.10.7.4.1

Cancel the common factor.

$v = \pm \frac{\sqrt{- \sqrt{e^{3 C} | x |} + x^{2}} \sqrt{3}}{x \cdot 3^{\frac{2}{2}}}$

Step 5.8.5.11.10.7.4.2

Rewrite the expression.

$v = \pm \frac{\sqrt{- \sqrt{e^{3 C} | x |} + x^{2}} \sqrt{3}}{x \cdot 3^{1}}$

Step 5.8.5.11.10.7.5

Evaluate the exponent.

$v = \pm \frac{\sqrt{- \sqrt{e^{3 C} | x |} + x^{2}} \sqrt{3}}{x \cdot 3}$

Step 5.8.5.11.11

Combine using the product rule for radicals.

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

Step 5.8.5.11.12

Simplify the expression.

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Step 5.8.5.11.12.1

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

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

Step 5.8.5.11.12.2

Reorder factors in $\pm \frac{\sqrt{(- \sqrt{e^{3 C} | x |} + x^{2}) \cdot 3}}{3 x}$ .

$v = \pm \frac{\sqrt{3 (- \sqrt{e^{3 C} | x |} + x^{2})}}{3 x}$

Step 5.8.5.12

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

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Step 5.8.5.12.1

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

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

Step 5.8.5.12.2

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

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

Step 5.8.5.12.3

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

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

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

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

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

Step 5.8.5.13

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Step 6

Simplify the constant of integration.

$v = \frac{\sqrt{3 (\sqrt{C | x |} + x^{2})}}{3 x}$

$v = - \frac{\sqrt{3 (\sqrt{C | x |} + x^{2})}}{3 x}$

$v = \frac{\sqrt{3 (- \sqrt{C | x |} + x^{2})}}{3 x}$

$v = - \frac{\sqrt{3 (- \sqrt{C | x |} + x^{2})}}{3 x}$