Calculus Examples

$\sqrt{y^{2} + 1} d x - x y d y = 0$

Step 1

Subtract $\sqrt{y^{2} + 1} d x$ from both sides of the equation.

$- x y d y = - \sqrt{y^{2} + 1} d x$

Step 2

Multiply both sides by $\frac{1}{x \sqrt{y^{2} + 1}}$ .

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

Step 3

Simplify.

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Step 3.1

Rewrite using the commutative property of multiplication.

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

Step 3.2

Cancel the common factor of $x$ .

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Step 3.2.1

Move the leading negative in $- \frac{1}{x \sqrt{y^{2} + 1}}$ into the numerator.

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

Step 3.2.2

Factor $x$ out of $x \sqrt{y^{2} + 1}$ .

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

Step 3.2.3

Factor $x$ out of $x y$ .

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

Step 3.2.4

Cancel the common factor.

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

Step 3.2.5

Rewrite the expression.

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

Step 3.3

Combine $\frac{- 1}{\sqrt{y^{2} + 1}}$ and $y$ .

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

Step 3.4

Move the negative in front of the fraction.

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

Step 3.5

Multiply $\frac{y}{\sqrt{y^{2} + 1}}$ by $\frac{\sqrt{y^{2} + 1}}{\sqrt{y^{2} + 1}}$ .

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

Step 3.6

Combine and simplify the denominator.

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Step 3.6.1

Multiply $\frac{y}{\sqrt{y^{2} + 1}}$ by $\frac{\sqrt{y^{2} + 1}}{\sqrt{y^{2} + 1}}$ .

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

Step 3.6.2

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

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

Step 3.6.3

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

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

Step 3.6.4

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

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

Step 3.6.5

Add $1$ and $1$ .

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

Step 3.6.6

Rewrite ${\sqrt{y^{2} + 1}}^{2}$ as $y^{2} + 1$ .

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Step 3.6.6.1

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

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

Step 3.6.6.2

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

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

Step 3.6.6.3

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

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

Step 3.6.6.4

Cancel the common factor of $2$ .

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Step 3.6.6.4.1

Cancel the common factor.

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

Step 3.6.6.4.2

Rewrite the expression.

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

Step 3.6.6.5

Simplify.

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

Step 3.7

Rewrite using the commutative property of multiplication.

$- \frac{y \sqrt{y^{2} + 1}}{y^{2} + 1} d y = - \frac{1}{x \sqrt{y^{2} + 1}} \sqrt{y^{2} + 1} d x$

Step 3.8

Cancel the common factor of $\sqrt{y^{2} + 1}$ .

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Step 3.8.1

Move the leading negative in $- \frac{1}{x \sqrt{y^{2} + 1}}$ into the numerator.

$- \frac{y \sqrt{y^{2} + 1}}{y^{2} + 1} d y = \frac{- 1}{x \sqrt{y^{2} + 1}} \sqrt{y^{2} + 1} d x$

Step 3.8.2

Factor $\sqrt{y^{2} + 1}$ out of $x \sqrt{y^{2} + 1}$ .

$- \frac{y \sqrt{y^{2} + 1}}{y^{2} + 1} d y = \frac{- 1}{\sqrt{y^{2} + 1} x} \sqrt{y^{2} + 1} d x$

Step 3.8.3

Cancel the common factor.

$- \frac{y \sqrt{y^{2} + 1}}{y^{2} + 1} d y = \frac{- 1}{\sqrt{y^{2} + 1} x} \sqrt{y^{2} + 1} d x$

Step 3.8.4

Rewrite the expression.

$- \frac{y \sqrt{y^{2} + 1}}{y^{2} + 1} d y = \frac{- 1}{x} d x$

Step 3.9

Move the negative in front of the fraction.

$- \frac{y \sqrt{y^{2} + 1}}{y^{2} + 1} d y = - \frac{1}{x} d x$

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

$\int - \frac{y \sqrt{y^{2} + 1}}{y^{2} + 1} d y = \int - \frac{1}{x} d x$

Step 4.2

Integrate the left side.

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Step 4.2.1

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

$- \int \frac{y \sqrt{y^{2} + 1}}{y^{2} + 1} d y = \int - \frac{1}{x} d x$

Step 4.2.2

Let $u = y^{2} + 1$ . Then $d u = 2 y d y$ , so $\frac{1}{2} d u = y d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.2.2.1

Let $u = y^{2} + 1$ . Find $\frac{d u}{d y}$ .

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Step 4.2.2.1.1

Differentiate $y^{2} + 1$ .

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

Step 4.2.2.1.2

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

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

Step 4.2.2.1.3

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

$2 y + \frac{d}{d y} [1]$

Step 4.2.2.1.4

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

$2 y + 0$

Step 4.2.2.1.5

Add $2 y$ and $0$ .

$2 y$

Step 4.2.2.2

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

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

Step 4.2.3

Simplify.

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Step 4.2.3.1

Multiply $\frac{\sqrt{u}}{u}$ by $\frac{1}{2}$ .

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

Step 4.2.3.2

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

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

Step 4.2.4

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

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

Step 4.2.5

Simplify the expression.

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Step 4.2.5.1

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

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

Step 4.2.5.2

Simplify.

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Step 4.2.5.2.1

Move $u^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 4.2.5.2.2

Multiply $u$ by $u^{- \frac{1}{2}}$ by adding the exponents.

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Step 4.2.5.2.2.1

Multiply $u$ by $u^{- \frac{1}{2}}$ .

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Step 4.2.5.2.2.1.1

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

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

Step 4.2.5.2.2.1.2

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

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

Step 4.2.5.2.2.2

Write $1$ as a fraction with a common denominator.

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

Step 4.2.5.2.2.3

Combine the numerators over the common denominator.

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

Step 4.2.5.2.2.4

Subtract $1$ from $2$ .

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

Step 4.2.5.3

Apply basic rules of exponents.

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Step 4.2.5.3.1

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

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

Step 4.2.5.3.2

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

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Step 4.2.5.3.2.1

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

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

Step 4.2.5.3.2.2

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

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

Step 4.2.5.3.2.3

Move the negative in front of the fraction.

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

Step 4.2.6

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

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

Step 4.2.7

Simplify.

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Step 4.2.7.1

Rewrite $- \frac{1}{2} (2 u^{\frac{1}{2}} + C_{1})$ as $- \frac{1}{2} \cdot 2 u^{\frac{1}{2}} + C_{1}$ .

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

Step 4.2.7.2

Simplify.

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Step 4.2.7.2.1

Multiply $2$ by $- 1$ .

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

Step 4.2.7.2.2

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

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

Step 4.2.7.2.3

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

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Step 4.2.7.2.3.1

Factor $2$ out of $- 2$ .

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

Step 4.2.7.2.3.2

Cancel the common factors.

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Step 4.2.7.2.3.2.1

Factor $2$ out of $2$ .

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

Step 4.2.7.2.3.2.2

Cancel the common factor.

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

Step 4.2.7.2.3.2.3

Rewrite the expression.

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

Step 4.2.7.2.3.2.4

Divide $- 1$ by $1$ .

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

Step 4.2.8

Replace all occurrences of $u$ with $y^{2} + 1$ .

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

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.

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

Step 4.3.2

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

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

Step 4.3.3

Simplify.

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

Step 4.4

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

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

Step 5

Solve for $y$ .

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

Simplify the left side.

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Step 5.1.2.1

Dividing two negative values results in a positive value.

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

Step 5.1.2.2

Divide ${(y^{2} + 1)}^{\frac{1}{2}}$ by $1$ .

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

Step 5.1.3

Simplify the right side.

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Step 5.1.3.1

Simplify each term.

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Step 5.1.3.1.1

Dividing two negative values results in a positive value.

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

Step 5.1.3.1.2

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

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

Step 5.1.3.1.3

Move the negative one from the denominator of $\frac{C}{- 1}$ .

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

Step 5.1.3.1.4

Rewrite $- 1 \cdot C$ as $- C$ .

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

Step 5.2

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

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

Step 5.3

Simplify the left side.

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Step 5.3.1

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

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Step 5.3.1.1

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

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Step 5.3.1.1.1

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

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

Step 5.3.1.1.2

Cancel the common factor of $2$ .

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Step 5.3.1.1.2.1

Cancel the common factor.

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

Step 5.3.1.1.2.2

Rewrite the expression.

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

Step 5.3.1.2

Simplify.

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

Step 5.4

Solve for $y$ .

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Step 5.4.1

Subtract $1$ from both sides of the equation.

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

Step 5.4.2

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

$y = \pm \sqrt{{(\ln (| x |) - C)}^{2} - 1}$

Step 5.4.3

Simplify $\pm \sqrt{{(\ln (| x |) - C)}^{2} - 1}$ .

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Step 5.4.3.1

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

$y = \pm \sqrt{{(\ln (| x |) - C)}^{2} - 1^{2}}$

Step 5.4.3.2

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

$y = \pm \sqrt{(\ln (| x |) - C + 1) (\ln (| x |) - C - 1)}$

Step 5.4.4

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

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Step 5.4.4.1

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

$y = \sqrt{(\ln (| x |) - C + 1) (\ln (| x |) - C - 1)}$

Step 5.4.4.2

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