Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Simplify.

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Step 3.1

Rewrite using the commutative property of multiplication.

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

Step 3.2

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

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

Step 3.3

Cancel the common factor of $x$ .

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Step 3.3.1

Factor $x$ out of $x v$ .

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 3.4

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

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

Step 3.5

Rewrite using the commutative property of multiplication.

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

Step 3.6

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

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Step 3.6.1

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

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

Step 3.6.2

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

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

Step 3.6.3

Cancel the common factor.

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

Step 3.6.4

Rewrite the expression.

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

Step 3.7

Move the negative in front of the fraction.

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

Integrate the left side.

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Step 4.2.1

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

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

Step 4.2.2

Let $u = 2 v^{2} - 1$ . Then $d u = 4 v d v$ , so $\frac{1}{4} d u = v d v$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.2.2.1

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

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Step 4.2.2.1.1

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

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

Step 4.2.2.1.2

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

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

Step 4.2.2.1.3

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

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Step 4.2.2.1.3.1

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

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

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 $n = 2$ .

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

Step 4.2.2.1.3.3

Multiply $2$ by $2$ .

$4 v + \frac{d}{d v} [- 1]$

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 $0$ .

$4 v + 0$

Step 4.2.2.1.4.2

Add $4 v$ and $0$ .

$4 v$

Step 4.2.2.2

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

$3 \int \frac{1}{u} \cdot \frac{1}{4} d u = \int - \frac{1}{x} d x$

Step 4.2.3

Simplify.

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Step 4.2.3.1

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

$3 \int \frac{1}{u \cdot 4} d u = \int - \frac{1}{x} d x$

Step 4.2.3.2

Move $4$ to the left of $u$ .

$3 \int \frac{1}{4 u} d u = \int - \frac{1}{x} d x$

Step 4.2.4

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$3 (\frac{1}{4} \int \frac{1}{u} d u) = \int - \frac{1}{x} d x$

Step 4.2.5

Combine $\frac{1}{4}$ and $3$ .

$\frac{3}{4} \int \frac{1}{u} d u = \int - \frac{1}{x} d x$

Step 4.2.6

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

$\frac{3}{4} (\ln (| u |) + C_{1}) = \int - \frac{1}{x} d x$

Step 4.2.7

Simplify.

$\frac{3}{4} \ln (| u |) + C_{1} = \int - \frac{1}{x} d x$

Step 4.2.8

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

$\frac{3}{4} \ln (| 2 v^{2} - 1 |) + C_{1} = \int - \frac{1}{x} d x$

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.

$\frac{3}{4} \ln (| 2 v^{2} - 1 |) + C_{1} = - \int \frac{1}{x} d x$

Step 4.3.2

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

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

Step 4.3.3

Simplify.

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

Step 4.4

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

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

Step 5

Solve for $v$ .

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Step 5.1

Multiply both sides of the equation by $\frac{4}{3}$ .

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

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

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Step 5.2.1.1.1

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

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

Step 5.2.1.1.2

Combine.

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

Step 5.2.1.1.3

Cancel the common factor of $4$ .

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Step 5.2.1.1.3.1

Cancel the common factor.

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

Step 5.2.1.1.3.2

Rewrite the expression.

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

Step 5.2.1.1.4

Cancel the common factor of $3$ .

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Step 5.2.1.1.4.1

Cancel the common factor.

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

Step 5.2.1.1.4.2

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

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

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $\frac{4}{3} (- \ln (| x |) + C)$ .

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Step 5.2.2.1.1

Apply the distributive property.

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

Step 5.2.2.1.2

Combine $\frac{4}{3}$ and $\ln (| x |)$ .

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

Step 5.2.2.1.3

Combine $\frac{4}{3}$ and $C$ .

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

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

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

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 $3$ .

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Step 5.3.3.1.1.1

Move the leading negative in $- \frac{4 \ln (| x |)}{3}$ into the numerator.

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

Step 5.3.3.1.1.2

Cancel the common factor.

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

Step 5.3.3.1.1.3

Rewrite the expression.

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

Step 5.3.3.1.2

Cancel the common factor of $3$ .

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Step 5.3.3.1.2.1

Cancel the common factor.

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

Step 5.3.3.1.2.2

Rewrite the expression.

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

Step 5.4

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

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

Step 5.5

Simplify the left side.

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Step 5.5.1

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

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Step 5.5.1.1

Simplify each term.

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Step 5.5.1.1.1

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

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

Step 5.5.1.1.2

Simplify $4 \ln (| x |)$ by moving $4$ inside the logarithm.

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

Step 5.5.1.1.3

Remove the absolute value in ${| x |}^{4}$ because exponentiations with even powers are always positive.

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

Step 5.5.1.2

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

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

Step 5.5.1.3

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

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

Step 5.6

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

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

Step 5.7

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

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

Step 5.8

Solve for $v$ .

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Step 5.8.1

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

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

Step 5.8.2

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

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Step 5.8.2.1

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

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

Step 5.8.2.2

Simplify the left side.

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Step 5.8.2.2.1

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

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Step 5.8.2.2.1.1

Cancel the common factor.

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

Step 5.8.2.2.1.2

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

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

Step 5.8.3

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

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

Step 5.8.4

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

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Step 5.8.4.1

Rewrite $\frac{e^{4 C}}{x^{4}}$ as ${(\frac{1}{x})}^{3} \frac{e^{4 C}}{x}$ .

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Step 5.8.4.1.1

Factor the perfect power $1^{3}$ out of $e^{4 C}$ .

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

Step 5.8.4.1.2

Factor the perfect power $x^{3}$ out of $x^{4}$ .

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

Step 5.8.4.1.3

Rearrange the fraction $\frac{1^{3} e^{4 C}}{x^{3} x}$ .

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

Step 5.8.4.2

Pull terms out from under the radical.

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

Step 5.8.4.3

Rewrite $\sqrt[3]{\frac{e^{4 C}}{x}}$ as $\frac{\sqrt[3]{e^{4 C}}}{\sqrt[3]{x}}$ .

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

Step 5.8.4.4

Combine.

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

Step 5.8.4.5

Multiply $\sqrt[3]{e^{4 C}}$ by $1$ .

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

Step 5.8.4.6

Multiply $\frac{\sqrt[3]{e^{4 C}}}{x \sqrt[3]{x}}$ by $\frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{2}}$ .

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

Step 5.8.4.7

Combine and simplify the denominator.

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Step 5.8.4.7.1

Multiply $\frac{\sqrt[3]{e^{4 C}}}{x \sqrt[3]{x}}$ by $\frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{2}}$ .

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

Step 5.8.4.7.2

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

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

Step 5.8.4.7.3

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

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

Step 5.8.4.7.4

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

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

Step 5.8.4.7.5

Add $1$ and $2$ .

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

Step 5.8.4.7.6

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

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Step 5.8.4.7.6.1

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

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

Step 5.8.4.7.6.2

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

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

Step 5.8.4.7.6.3

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

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

Step 5.8.4.7.6.4

Cancel the common factor of $3$ .

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Step 5.8.4.7.6.4.1

Cancel the common factor.

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

Step 5.8.4.7.6.4.2

Rewrite the expression.

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

Step 5.8.4.7.6.5

Simplify.

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

Step 5.8.4.8

Multiply $x$ by $x$ .

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

Step 5.8.4.9

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

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

Step 5.8.4.10

Combine using the product rule for radicals.

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

Step 5.8.4.11

Reorder factors in $\frac{\sqrt[3]{e^{4 C} x^{2}}}{x^{2}}$ .

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

Step 5.8.5

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

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

Step 5.8.6

Add $1$ to both sides of the equation.

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

Step 5.8.7

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

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Step 5.8.7.1

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

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

Step 5.8.7.2

Simplify the left side.

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Step 5.8.7.2.1

Cancel the common factor of $2$ .

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Step 5.8.7.2.1.1

Cancel the common factor.

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

Step 5.8.7.2.1.2

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

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

Step 5.8.8

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

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

Step 5.8.9

Simplify $\pm \sqrt{\frac{\pm \frac{\sqrt[3]{x^{2} e^{4 C}}}{x^{2}}}{2} + \frac{1}{2}}$ .

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Step 5.8.9.1

Combine the numerators over the common denominator.

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

Step 5.8.9.2

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

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

Step 5.8.9.3

Multiply $\frac{\sqrt{\pm \frac{\sqrt[3]{x^{2} e^{4 C}}}{x^{2}} + 1}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 5.8.9.4

Combine and simplify the denominator.

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Step 5.8.9.4.1

Multiply $\frac{\sqrt{\pm \frac{\sqrt[3]{x^{2} e^{4 C}}}{x^{2}} + 1}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 5.8.9.4.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 5.8.9.4.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 5.8.9.4.4

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

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

Step 5.8.9.4.5

Add $1$ and $1$ .

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

Step 5.8.9.4.6

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

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Step 5.8.9.4.6.1

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

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

Step 5.8.9.4.6.2

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

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

Step 5.8.9.4.6.3

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

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

Step 5.8.9.4.6.4

Cancel the common factor of $2$ .

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Step 5.8.9.4.6.4.1

Cancel the common factor.

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

Step 5.8.9.4.6.4.2

Rewrite the expression.

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

Step 5.8.9.4.6.5

Evaluate the exponent.

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

Step 5.8.9.5

Combine using the product rule for radicals.

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

Step 6

Simplify the constant of integration.

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