Calculus Examples

$x^{2} \frac{d w}{d x} = \sqrt{w} (2 x + 4)$

Step 1

Separate the variables.

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Step 1.1

Divide each term in $x^{2} \frac{d w}{d x} = \sqrt{w} (2 x + 4)$ by $x^{2}$ and simplify.

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Step 1.1.1

Divide each term in $x^{2} \frac{d w}{d x} = \sqrt{w} (2 x + 4)$ by $x^{2}$ .

$\frac{x^{2} \frac{d w}{d x}}{x^{2}} = \frac{\sqrt{w} (2 x + 4)}{x^{2}}$

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of $x^{2}$ .

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Step 1.1.2.1.1

Cancel the common factor.

$\frac{x^{2} \frac{d w}{d x}}{x^{2}} = \frac{\sqrt{w} (2 x + 4)}{x^{2}}$

Step 1.1.2.1.2

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

$\frac{d w}{d x} = \frac{\sqrt{w} (2 x + 4)}{x^{2}}$

Step 1.1.3

Simplify the right side.

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Step 1.1.3.1

Factor $2$ out of $2 x + 4$ .

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Step 1.1.3.1.1

Factor $2$ out of $2 x$ .

$\frac{d w}{d x} = \frac{\sqrt{w} (2 (x) + 4)}{x^{2}}$

Step 1.1.3.1.2

Factor $2$ out of $4$ .

$\frac{d w}{d x} = \frac{\sqrt{w} (2 x + 2 \cdot 2)}{x^{2}}$

Step 1.1.3.1.3

Factor $2$ out of $2 x + 2 \cdot 2$ .

$\frac{d w}{d x} = \frac{\sqrt{w} (2 (x + 2))}{x^{2}}$

$\frac{d w}{d x} = \frac{\sqrt{w} \cdot 2 (x + 2)}{x^{2}}$

Step 1.1.3.2

Move $2$ to the left of $\sqrt{w}$ .

$\frac{d w}{d x} = \frac{2 \sqrt{w} (x + 2)}{x^{2}}$

Step 1.2

Regroup factors.

$\frac{d w}{d x} = 2 \sqrt{w} \frac{x + 2}{x^{2}}$

Step 1.3

Multiply both sides by $\frac{1}{\sqrt{w}}$ .

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

Step 1.4

Simplify.

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Step 1.4.1

Rewrite using the commutative property of multiplication.

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

Step 1.4.2

Multiply $\frac{1}{\sqrt{w}}$ by $\frac{\sqrt{w}}{\sqrt{w}}$ .

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

Step 1.4.3

Combine and simplify the denominator.

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Step 1.4.3.1

Multiply $\frac{1}{\sqrt{w}}$ by $\frac{\sqrt{w}}{\sqrt{w}}$ .

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

Step 1.4.3.2

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

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

Step 1.4.3.3

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

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

Step 1.4.3.4

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

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

Step 1.4.3.5

Add $1$ and $1$ .

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

Step 1.4.3.6

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

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Step 1.4.3.6.1

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

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

Step 1.4.3.6.2

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

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

Step 1.4.3.6.3

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

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

Step 1.4.3.6.4

Cancel the common factor of $2$ .

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Step 1.4.3.6.4.1

Cancel the common factor.

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

Step 1.4.3.6.4.2

Rewrite the expression.

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

Step 1.4.3.6.5

Simplify.

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

Step 1.4.4

Combine $2$ and $\frac{\sqrt{w}}{w}$ .

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

Step 1.4.5

Combine $\sqrt{w}$ and $\frac{x + 2}{x^{2}}$ .

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

Step 1.4.6

Combine.

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

Step 1.4.7

Simplify the numerator.

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Step 1.4.7.1

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

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

Step 1.4.7.2

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

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

Step 1.4.7.3

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

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

Step 1.4.7.4

Add $1$ and $1$ .

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

Step 1.4.8

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

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Step 1.4.8.1

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

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

Step 1.4.8.2

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

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

Step 1.4.8.3

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

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

Step 1.4.8.4

Cancel the common factor of $2$ .

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Step 1.4.8.4.1

Cancel the common factor.

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

Step 1.4.8.4.2

Rewrite the expression.

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

Step 1.4.8.5

Simplify.

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

Step 1.4.9

Cancel the common factor of $w$ .

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Step 1.4.9.1

Cancel the common factor.

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

Step 1.4.9.2

Rewrite the expression.

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

Step 1.5

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Apply basic rules of exponents.

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Step 2.2.1.1

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

$\int \frac{1}{w^{\frac{1}{2}}} d w = \int \frac{2 (x + 2)}{x^{2}} d x$

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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Step 2.2.1.3.1

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

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

Step 2.2.1.3.2

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

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

Step 2.2.1.3.3

Move the negative in front of the fraction.

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

Step 2.2.2

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

$2 w^{\frac{1}{2}} + C_{1} = \int \frac{2 (x + 2)}{x^{2}} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 w^{\frac{1}{2}} + C_{1} = 2 \int \frac{x + 2}{x^{2}} d x$

Step 2.3.2

Apply basic rules of exponents.

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Step 2.3.2.1

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

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

Step 2.3.2.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 2.3.2.2.1

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

$2 w^{\frac{1}{2}} + C_{1} = 2 \int (x + 2) x^{2 \cdot - 1} d x$

Step 2.3.2.2.2

Multiply $2$ by $- 1$ .

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

Step 2.3.3

Multiply $(x + 2) x^{- 2}$ .

$2 w^{\frac{1}{2}} + C_{1} = 2 \int x \cdot x^{- 2} + 2 x^{- 2} d x$

Step 2.3.4

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

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Step 2.3.4.1

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

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Step 2.3.4.1.1

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

$2 w^{\frac{1}{2}} + C_{1} = 2 \int x^{1} x^{- 2} + 2 x^{- 2} d x$

Step 2.3.4.1.2

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

$2 w^{\frac{1}{2}} + C_{1} = 2 \int x^{1 - 2} + 2 x^{- 2} d x$

Step 2.3.4.2

Subtract $2$ from $1$ .

$2 w^{\frac{1}{2}} + C_{1} = 2 \int x^{- 1} + 2 x^{- 2} d x$

Step 2.3.5

Split the single integral into multiple integrals.

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

Step 2.3.6

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

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

Step 2.3.7

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 2.3.8

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

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

Step 2.3.9

Simplify.

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

Step 2.4

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

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

Step 3

Solve for $w$ .

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Step 3.1

Divide each term in $2 w^{\frac{1}{2}} = 2 (\ln (| x |) - \frac{2}{x}) + C$ by $2$ and simplify.

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Step 3.1.1

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

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

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor.

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

Step 3.1.2.2

Divide $w^{\frac{1}{2}}$ by $1$ .

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

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Combine the numerators over the common denominator.

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

Step 3.1.3.2

Simplify the numerator.

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Step 3.1.3.2.1

Apply the distributive property.

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

Step 3.1.3.2.2

Simplify $2 \ln (| x |)$ by moving $2$ inside the logarithm.

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

Step 3.1.3.2.3

Multiply $2 (- \frac{2}{x})$ .

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Step 3.1.3.2.3.1

Multiply $- 1$ by $2$ .

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

Step 3.1.3.2.3.2

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

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

Step 3.1.3.2.3.3

Multiply $- 2$ by $2$ .

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

Step 3.1.3.2.4

Simplify each term.

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Step 3.1.3.2.4.1

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

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

Step 3.1.3.2.4.2

Move the negative in front of the fraction.

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

Step 3.1.3.2.5

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

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

Step 3.1.3.2.6

Combine the numerators over the common denominator.

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

Step 3.1.3.2.7

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

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

Step 3.1.3.2.8

Combine the numerators over the common denominator.

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

Step 3.1.3.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.1.3.4

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

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

Step 3.1.3.5

Simplify the expression.

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Step 3.1.3.5.1

Move $2$ to the left of $x$ .

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

Step 3.1.3.5.2

Reorder factors in $\frac{\ln (x^{2}) x - 4 + C x}{2 x}$ .

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

Step 3.2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Step 3.3

Simplify the exponent.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Simplify ${(w^{\frac{1}{2}})}^{2}$ .

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Step 3.3.1.1.1

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

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Step 3.3.1.1.1.1

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

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

Step 3.3.1.1.1.2

Cancel the common factor of $2$ .

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Step 3.3.1.1.1.2.1

Cancel the common factor.

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

Step 3.3.1.1.1.2.2

Rewrite the expression.

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

Step 3.3.1.1.2

Simplify.

$w = {(\frac{x \ln (x^{2}) - 4 + C x}{2 x})}^{2}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

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

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Step 3.3.2.1.1

Split the fraction $\frac{x \ln (x^{2}) - 4 + C x}{2 x}$ into two fractions.

$w = {(\frac{x \ln (x^{2}) - 4}{2 x} + \frac{C x}{2 x})}^{2}$

Step 3.3.2.1.2

Simplify each term.

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Step 3.3.2.1.2.1

Split the fraction $\frac{x \ln (x^{2}) - 4}{2 x}$ into two fractions.

$w = {(\frac{x \ln (x^{2})}{2 x} + \frac{- 4}{2 x} + \frac{C x}{2 x})}^{2}$

Step 3.3.2.1.2.2

Simplify each term.

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Step 3.3.2.1.2.2.1

Expand $\ln (x^{2})$ by moving $2$ outside the logarithm.

$w = {(\frac{x (2 \ln (x))}{2 x} + \frac{- 4}{2 x} + \frac{C x}{2 x})}^{2}$

Step 3.3.2.1.2.2.2

Cancel the common factor of $x$ .

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Step 3.3.2.1.2.2.2.1

Cancel the common factor.

$w = {(\frac{x (2 \ln (x))}{2 x} + \frac{- 4}{2 x} + \frac{C x}{2 x})}^{2}$

Step 3.3.2.1.2.2.2.2

Rewrite the expression.

$w = {(\frac{2 \ln (x)}{2} + \frac{- 4}{2 x} + \frac{C x}{2 x})}^{2}$

Step 3.3.2.1.2.2.3

Cancel the common factor of $2$ .

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Step 3.3.2.1.2.2.3.1

Cancel the common factor.

$w = {(\frac{2 \ln (x)}{2} + \frac{- 4}{2 x} + \frac{C x}{2 x})}^{2}$

Step 3.3.2.1.2.2.3.2

Divide $\ln (x)$ by $1$ .

$w = {(\ln (x) + \frac{- 4}{2 x} + \frac{C x}{2 x})}^{2}$

Step 3.3.2.1.2.2.4

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

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Step 3.3.2.1.2.2.4.1

Factor $2$ out of $- 4$ .

$w = {(\ln (x) + \frac{2 \cdot - 2}{2 x} + \frac{C x}{2 x})}^{2}$

Step 3.3.2.1.2.2.4.2

Cancel the common factors.

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Step 3.3.2.1.2.2.4.2.1

Factor $2$ out of $2 x$ .

$w = {(\ln (x) + \frac{2 \cdot - 2}{2 (x)} + \frac{C x}{2 x})}^{2}$

Step 3.3.2.1.2.2.4.2.2

Cancel the common factor.

$w = {(\ln (x) + \frac{2 \cdot - 2}{2 x} + \frac{C x}{2 x})}^{2}$

Step 3.3.2.1.2.2.4.2.3

Rewrite the expression.

$w = {(\ln (x) + \frac{- 2}{x} + \frac{C x}{2 x})}^{2}$

Step 3.3.2.1.2.2.5

Move the negative in front of the fraction.

$w = {(\ln (x) - \frac{2}{x} + \frac{C x}{2 x})}^{2}$

Step 3.3.2.1.2.3

Cancel the common factor of $x$ .

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Step 3.3.2.1.2.3.1

Cancel the common factor.

$w = {(\ln (x) - \frac{2}{x} + \frac{C x}{2 x})}^{2}$

Step 3.3.2.1.2.3.2

Rewrite the expression.

$w = {(\ln (x) - \frac{2}{x} + \frac{C}{2})}^{2}$

Step 4

Simplify the constant of integration.

$w = {(\ln (x) - \frac{2}{x} + C)}^{2}$