Calculus Examples

$x^{2} \frac{d w}{d x} = \sqrt{w} (9 x + 3)$

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Divide each term in $x^{2} \frac{d w}{d x} = \sqrt{w} (9 x + 3)$ by $x^{2}$ and simplify.

Tap for more steps...

Step 1.1.1

Divide each term in $x^{2} \frac{d w}{d x} = \sqrt{w} (9 x + 3)$ by $x^{2}$ .

$\frac{x^{2} \frac{d w}{d x}}{x^{2}} = \frac{\sqrt{w} (9 x + 3)}{x^{2}}$

Step 1.1.2

Simplify the left side.

Tap for more steps...

Step 1.1.2.1

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

Tap for more steps...

Step 1.1.2.1.1

Cancel the common factor.

$\frac{x^{2} \frac{d w}{d x}}{x^{2}} = \frac{\sqrt{w} (9 x + 3)}{x^{2}}$

Step 1.1.2.1.2

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

$\frac{d w}{d x} = \frac{\sqrt{w} (9 x + 3)}{x^{2}}$

Step 1.1.3

Simplify the right side.

Tap for more steps...

Step 1.1.3.1

Factor $3$ out of $9 x + 3$ .

Tap for more steps...

Step 1.1.3.1.1

Factor $3$ out of $9 x$ .

$\frac{d w}{d x} = \frac{\sqrt{w} (3 (3 x) + 3)}{x^{2}}$

Step 1.1.3.1.2

Factor $3$ out of $3$ .

$\frac{d w}{d x} = \frac{\sqrt{w} (3 (3 x) + 3 (1))}{x^{2}}$

Step 1.1.3.1.3

Factor $3$ out of $3 (3 x) + 3 (1)$ .

$\frac{d w}{d x} = \frac{\sqrt{w} (3 (3 x + 1))}{x^{2}}$

$\frac{d w}{d x} = \frac{\sqrt{w} \cdot 3 (3 x + 1)}{x^{2}}$

Step 1.1.3.2

Move $3$ to the left of $\sqrt{w}$ .

$\frac{d w}{d x} = \frac{3 \sqrt{w} (3 x + 1)}{x^{2}}$

Step 1.2

Regroup factors.

$\frac{d w}{d x} = 3 \sqrt{w} \frac{3 x + 1}{x^{2}}$

Step 1.3

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

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = \frac{1}{\sqrt{w}} (3 \sqrt{w} \frac{3 x + 1}{x^{2}})$

Step 1.4

Simplify.

Tap for more steps...

Step 1.4.1

Rewrite using the commutative property of multiplication.

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = 3 \frac{1}{\sqrt{w}} (\sqrt{w} \frac{3 x + 1}{x^{2}})$

Step 1.4.2

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

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = 3 (\frac{1}{\sqrt{w}} \cdot \frac{\sqrt{w}}{\sqrt{w}}) (\sqrt{w} \frac{3 x + 1}{x^{2}})$

Step 1.4.3

Combine and simplify the denominator.

Tap for more steps...

Step 1.4.3.1

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

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = 3 \frac{\sqrt{w}}{\sqrt{w} \sqrt{w}} (\sqrt{w} \frac{3 x + 1}{x^{2}})$

Step 1.4.3.2

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

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = 3 \frac{\sqrt{w}}{{\sqrt{w}}^{1} \sqrt{w}} (\sqrt{w} \frac{3 x + 1}{x^{2}})$

Step 1.4.3.3

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

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = 3 \frac{\sqrt{w}}{{\sqrt{w}}^{1} {\sqrt{w}}^{1}} (\sqrt{w} \frac{3 x + 1}{x^{2}})$

Step 1.4.3.4

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

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = 3 \frac{\sqrt{w}}{{\sqrt{w}}^{1 + 1}} (\sqrt{w} \frac{3 x + 1}{x^{2}})$

Step 1.4.3.5

Add $1$ and $1$ .

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = 3 \frac{\sqrt{w}}{{\sqrt{w}}^{2}} (\sqrt{w} \frac{3 x + 1}{x^{2}})$

Step 1.4.3.6

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

Tap for more steps...

Step 1.4.3.6.1

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

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = 3 \frac{\sqrt{w}}{{(w^{\frac{1}{2}})}^{2}} (\sqrt{w} \frac{3 x + 1}{x^{2}})$

Step 1.4.3.6.2

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

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = 3 \frac{\sqrt{w}}{w^{\frac{1}{2} \cdot 2}} (\sqrt{w} \frac{3 x + 1}{x^{2}})$

Step 1.4.3.6.3

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

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = 3 \frac{\sqrt{w}}{w^{\frac{2}{2}}} (\sqrt{w} \frac{3 x + 1}{x^{2}})$

Step 1.4.3.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.4.3.6.4.1

Cancel the common factor.

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = 3 \frac{\sqrt{w}}{w^{\frac{2}{2}}} (\sqrt{w} \frac{3 x + 1}{x^{2}})$

Step 1.4.3.6.4.2

Rewrite the expression.

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = 3 \frac{\sqrt{w}}{w^{1}} (\sqrt{w} \frac{3 x + 1}{x^{2}})$

Step 1.4.3.6.5

Simplify.

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = 3 \frac{\sqrt{w}}{w} (\sqrt{w} \frac{3 x + 1}{x^{2}})$

Step 1.4.4

Combine $3$ and $\frac{\sqrt{w}}{w}$ .

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = \frac{3 \sqrt{w}}{w} (\sqrt{w} \frac{3 x + 1}{x^{2}})$

Step 1.4.5

Combine $\sqrt{w}$ and $\frac{3 x + 1}{x^{2}}$ .

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = \frac{3 \sqrt{w}}{w} \cdot \frac{\sqrt{w} (3 x + 1)}{x^{2}}$

Step 1.4.6

Combine.

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = \frac{3 \sqrt{w} (\sqrt{w} (3 x + 1))}{w x^{2}}$

Step 1.4.7

Simplify the numerator.

Tap for more steps...

Step 1.4.7.1

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

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = \frac{3 ({\sqrt{w}}^{1} \sqrt{w}) (3 x + 1)}{w x^{2}}$

Step 1.4.7.2

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

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = \frac{3 ({\sqrt{w}}^{1} {\sqrt{w}}^{1}) (3 x + 1)}{w x^{2}}$

Step 1.4.7.3

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

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = \frac{3 {\sqrt{w}}^{1 + 1} (3 x + 1)}{w x^{2}}$

Step 1.4.7.4

Add $1$ and $1$ .

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = \frac{3 {\sqrt{w}}^{2} (3 x + 1)}{w x^{2}}$

Step 1.4.8

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

Tap for more steps...

Step 1.4.8.1

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

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = \frac{3 {(w^{\frac{1}{2}})}^{2} (3 x + 1)}{w x^{2}}$

Step 1.4.8.2

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

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = \frac{3 w^{\frac{1}{2} \cdot 2} (3 x + 1)}{w x^{2}}$

Step 1.4.8.3

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

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = \frac{3 w^{\frac{2}{2}} (3 x + 1)}{w x^{2}}$

Step 1.4.8.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.4.8.4.1

Cancel the common factor.

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = \frac{3 w^{\frac{2}{2}} (3 x + 1)}{w x^{2}}$

Step 1.4.8.4.2

Rewrite the expression.

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = \frac{3 w^{1} (3 x + 1)}{w x^{2}}$

Step 1.4.8.5

Simplify.

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = \frac{3 w (3 x + 1)}{w x^{2}}$

Step 1.4.9

Cancel the common factor of $w$ .

Tap for more steps...

Step 1.4.9.1

Cancel the common factor.

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = \frac{3 w (3 x + 1)}{w x^{2}}$

Step 1.4.9.2

Rewrite the expression.

$\frac{1}{\sqrt{w}} \frac{d w}{d x} = \frac{3 (3 x + 1)}{x^{2}}$

Step 1.5

Rewrite the equation.

$\frac{1}{\sqrt{w}} d w = \frac{3 (3 x + 1)}{x^{2}} d x$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int \frac{1}{\sqrt{w}} d w = \int \frac{3 (3 x + 1)}{x^{2}} d x$

Step 2.2

Integrate the left side.

Tap for more steps...

Step 2.2.1

Apply basic rules of exponents.

Tap for more steps...

Step 2.2.1.1

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

$\int \frac{1}{w^{\frac{1}{2}}} d w = \int \frac{3 (3 x + 1)}{x^{2}} d x$

Step 2.2.1.2

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

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

Step 2.2.1.3

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

Tap for more steps...

Step 2.2.1.3.1

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

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

Step 2.2.1.3.2

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

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

Step 2.2.1.3.3

Move the negative in front of the fraction.

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

Step 2.2.2

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

$2 w^{\frac{1}{2}} + C_{1} = \int \frac{3 (3 x + 1)}{x^{2}} d x$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

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

$2 w^{\frac{1}{2}} + C_{1} = 3 \int \frac{3 x + 1}{x^{2}} d x$

Step 2.3.2

Apply basic rules of exponents.

Tap for more steps...

Step 2.3.2.1

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

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

Step 2.3.2.2

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

Tap for more steps...

Step 2.3.2.2.1

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

$2 w^{\frac{1}{2}} + C_{1} = 3 \int (3 x + 1) x^{2 \cdot - 1} d x$

Step 2.3.2.2.2

Multiply $2$ by $- 1$ .

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

Step 2.3.3

Multiply $(3 x + 1) x^{- 2}$ .

$2 w^{\frac{1}{2}} + C_{1} = 3 \int 3 x \cdot x^{- 2} + 1 x^{- 2} d x$

Step 2.3.4

Simplify.

Tap for more steps...

Step 2.3.4.1

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

Tap for more steps...

Step 2.3.4.1.1

Move $x^{- 2}$ .

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

Step 2.3.4.1.2

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

Tap for more steps...

Step 2.3.4.1.2.1

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

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

Step 2.3.4.1.2.2

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

$2 w^{\frac{1}{2}} + C_{1} = 3 \int 3 x^{- 2 + 1} + 1 x^{- 2} d x$

Step 2.3.4.1.3

Add $- 2$ and $1$ .

$2 w^{\frac{1}{2}} + C_{1} = 3 \int 3 x^{- 1} + 1 x^{- 2} d x$

Step 2.3.4.2

Multiply $x^{- 2}$ by $1$ .

$2 w^{\frac{1}{2}} + C_{1} = 3 \int 3 x^{- 1} + x^{- 2} d x$

Step 2.3.5

Split the single integral into multiple integrals.

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

Step 2.3.6

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

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

Step 2.3.7

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

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

Step 2.3.8

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

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

Step 2.3.9

Simplify.

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

Step 2.4

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

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

Step 3

Solve for $w$ .

Tap for more steps...

Step 3.1

Divide each term in $2 w^{\frac{1}{2}} = 3 (3 \ln (| x |) - \frac{1}{x}) + C$ by $2$ and simplify.

Tap for more steps...

Step 3.1.1

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

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

Step 3.1.2

Simplify the left side.

Tap for more steps...

Step 3.1.2.1

Cancel the common factor.

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

Step 3.1.2.2

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

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

Step 3.1.3

Simplify the right side.

Tap for more steps...

Step 3.1.3.1

Simplify each term.

Tap for more steps...

Step 3.1.3.1.1

Simplify the numerator.

Tap for more steps...

Step 3.1.3.1.1.1

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

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

Step 3.1.3.1.1.2

To write $\ln ({| x |}^{3})$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 3.1.3.1.1.3

Combine the numerators over the common denominator.

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

Step 3.1.3.1.2

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

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

Step 3.1.3.1.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.1.3.1.4

Combine.

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

Step 3.1.3.1.5

Multiply $3$ by $1$ .

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

Step 3.1.3.1.6

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

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

Step 3.1.3.2

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

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

Step 3.1.3.3

Simplify terms.

Tap for more steps...

Step 3.1.3.3.1

Multiply $\frac{C}{2}$ by $\frac{x}{x}$ .

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

Step 3.1.3.3.2

Combine the numerators over the common denominator.

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

Step 3.1.3.4

Simplify the numerator.

Tap for more steps...

Step 3.1.3.4.1

Apply the distributive property.

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

Step 3.1.3.4.2

Multiply $3 (\ln ({| x |}^{3}) x)$ .

Tap for more steps...

Step 3.1.3.4.2.1

Reorder $\ln ({| x |}^{3})$ and $3$ .

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

Step 3.1.3.4.2.2

Simplify $3 \cdot \ln ({| x |}^{3})$ by moving $3$ inside the logarithm.

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

Step 3.1.3.4.3

Multiply $3$ by $- 1$ .

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

Step 3.1.3.4.4

Multiply the exponents in ${({| x |}^{3})}^{3}$ .

Tap for more steps...

Step 3.1.3.4.4.1

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

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

Step 3.1.3.4.4.2

Multiply $3$ by $3$ .

$w^{\frac{1}{2}} = \frac{\ln ({| x |}^{9}) x - 3 + C x}{2 x}$

Step 3.1.3.5

Reorder factors in $\frac{\ln ({| x |}^{9}) x - 3 + C x}{2 x}$ .

$w^{\frac{1}{2}} = \frac{x \ln ({| x |}^{9}) - 3 + C x}{2 x}$

Step 3.2

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

${(w^{\frac{1}{2}})}^{2} = {(\frac{x \ln ({| x |}^{9}) - 3 + C x}{2 x})}^{2}$

Step 3.3

Simplify the exponent.

Tap for more steps...

Step 3.3.1

Simplify the left side.

Tap for more steps...

Step 3.3.1.1

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

Tap for more steps...

Step 3.3.1.1.1

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

Tap for more steps...

Step 3.3.1.1.1.1

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

$w^{\frac{1}{2} \cdot 2} = {(\frac{x \ln ({| x |}^{9}) - 3 + C x}{2 x})}^{2}$

Step 3.3.1.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.3.1.1.1.2.1

Cancel the common factor.

$w^{\frac{1}{2} \cdot 2} = {(\frac{x \ln ({| x |}^{9}) - 3 + C x}{2 x})}^{2}$

Step 3.3.1.1.1.2.2

Rewrite the expression.

$w^{1} = {(\frac{x \ln ({| x |}^{9}) - 3 + C x}{2 x})}^{2}$

Step 3.3.1.1.2

Simplify.

$w = {(\frac{x \ln ({| x |}^{9}) - 3 + C x}{2 x})}^{2}$

Step 3.3.2

Simplify the right side.

Tap for more steps...

Step 3.3.2.1

Simplify ${(\frac{x \ln ({| x |}^{9}) - 3 + C x}{2 x})}^{2}$ .

Tap for more steps...

Step 3.3.2.1.1

Split the fraction $\frac{x \ln ({| x |}^{9}) - 3 + C x}{2 x}$ into two fractions.

$w = {(\frac{x \ln ({| x |}^{9}) - 3}{2 x} + \frac{C x}{2 x})}^{2}$

Step 3.3.2.1.2

Simplify each term.

Tap for more steps...

Step 3.3.2.1.2.1

Split the fraction $\frac{x \ln ({| x |}^{9}) - 3}{2 x}$ into two fractions.

$w = {(\frac{x \ln ({| x |}^{9})}{2 x} + \frac{- 3}{2 x} + \frac{C x}{2 x})}^{2}$

Step 3.3.2.1.2.2

Simplify each term.

Tap for more steps...

Step 3.3.2.1.2.2.1

Expand $\ln ({| x |}^{9})$ by moving $9$ outside the logarithm.

$w = {(\frac{x (9 \ln (| x |))}{2 x} + \frac{- 3}{2 x} + \frac{C x}{2 x})}^{2}$

Step 3.3.2.1.2.2.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 3.3.2.1.2.2.2.1

Cancel the common factor.

$w = {(\frac{x (9 \ln (| x |))}{2 x} + \frac{- 3}{2 x} + \frac{C x}{2 x})}^{2}$

Step 3.3.2.1.2.2.2.2

Rewrite the expression.

$w = {(\frac{9 \ln (| x |)}{2} + \frac{- 3}{2 x} + \frac{C x}{2 x})}^{2}$

Step 3.3.2.1.2.2.3

Rewrite $\frac{9 \ln (| x |)}{2}$ as $\frac{9}{2} \ln (| x |)$ .

$w = {(\frac{9}{2} \ln (| x |) + \frac{- 3}{2 x} + \frac{C x}{2 x})}^{2}$

Step 3.3.2.1.2.2.4

Simplify $\frac{9}{2} \ln (| x |)$ by moving $\frac{9}{2}$ inside the logarithm.

$w = {(\ln ({| x |}^{\frac{9}{2}}) + \frac{- 3}{2 x} + \frac{C x}{2 x})}^{2}$

Step 3.3.2.1.2.2.5

Move the negative in front of the fraction.

$w = {(\ln ({| x |}^{\frac{9}{2}}) - \frac{3}{2 x} + \frac{C x}{2 x})}^{2}$

Step 3.3.2.1.2.3

Cancel the common factor of $x$ .

Tap for more steps...

Step 3.3.2.1.2.3.1

Cancel the common factor.

$w = {(\ln ({| x |}^{\frac{9}{2}}) - \frac{3}{2 x} + \frac{C x}{2 x})}^{2}$

Step 3.3.2.1.2.3.2

Rewrite the expression.

$w = {(\ln ({| x |}^{\frac{9}{2}}) - \frac{3}{2 x} + \frac{C}{2})}^{2}$

Step 4

Simplify the constant of integration.

$w = {(\ln ({| x |}^{\frac{9}{2}}) - \frac{3}{2 x} + C)}^{2}$