Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Simplify.

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Step 3.1

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

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Step 3.1.1

Factor $x^{2}$ out of $x^{2} y$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Multiply $\frac{1}{y}$ by $y + 1$ .

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

Step 3.3

Rewrite using the commutative property of multiplication.

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

Step 3.4

Cancel the common factor of $y$ .

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Step 3.4.1

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

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

Step 3.4.2

Factor $y$ out of $x^{2} y$ .

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

Step 3.4.3

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

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

Step 3.4.4

Cancel the common factor.

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

Step 3.4.5

Rewrite the expression.

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

Step 3.5

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

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

Step 3.6

Move the negative in front of the fraction.

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

Integrate the left side.

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Step 4.2.1

Split the fraction into multiple fractions.

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

Step 4.2.2

Split the single integral into multiple integrals.

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

Step 4.2.3

Cancel the common factor of $y$ .

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Step 4.2.3.1

Cancel the common factor.

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

Step 4.2.3.2

Rewrite the expression.

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

Step 4.2.4

Apply the constant rule.

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

Step 4.2.5

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

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

Step 4.2.6

Simplify.

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

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.

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

Step 4.3.2

Apply basic rules of exponents.

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Step 4.3.2.1

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

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

Step 4.3.2.2

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

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Step 4.3.2.2.1

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

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

Step 4.3.2.2.2

Multiply $2$ by $- 1$ .

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

Step 4.3.3

Let $u_{1} = x - 1$ . Then $d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 4.3.3.1

Let $u_{1} = x - 1$ . Find $\frac{d u_{1}}{d x}$ .

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Step 4.3.3.1.1

Differentiate $x - 1$ .

$\frac{d}{d x} [x - 1]$

Step 4.3.3.1.2

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

$\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 4.3.3.1.3

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

$1 + \frac{d}{d x} [- 1]$

Step 4.3.3.1.4

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

$1 + 0$

Step 4.3.3.1.5

Add $1$ and $0$ .

$1$

Step 4.3.3.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$y + \ln (| y |) + C_{3} = - \int {u_{1}}^{2} {(u_{1} + 1)}^{- 2} d u_{1}$

Step 4.3.4

Let $u_{2} = u_{1} + 1$ . Then $d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 4.3.4.1

Let $u_{2} = u_{1} + 1$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 4.3.4.1.1

Differentiate $u_{1} + 1$ .

$\frac{d}{d u_{1}} [u_{1} + 1]$

Step 4.3.4.1.2

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

$\frac{d}{d u_{1}} [u_{1}] + \frac{d}{d u_{1}} [1]$

Step 4.3.4.1.3

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

$1 + \frac{d}{d u_{1}} [1]$

Step 4.3.4.1.4

Since $1$ is constant with respect to $u_{1}$ , the derivative of $1$ with respect to $u_{1}$ is $0$ .

$1 + 0$

Step 4.3.4.1.5

Add $1$ and $0$ .

$1$

Step 4.3.4.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$y + \ln (| y |) + C_{3} = - \int {(u_{2} - 1)}^{2} {u_{2}}^{- 2} d u_{2}$

Step 4.3.5

Let $u_{3} = {u_{2}}^{2}$ . Then $d u_{3} = 2 u_{2} d u_{2}$ , so $\frac{1}{2} d u_{3} = u_{2} d u_{2}$ . Rewrite using $u_{3}$ and $d$ $u_{3}$ .

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Step 4.3.5.1

Let $u_{3} = {u_{2}}^{2}$ . Find $\frac{d u_{3}}{d u_{2}}$ .

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Step 4.3.5.1.1

Differentiate ${u_{2}}^{2}$ .

$\frac{d}{d u_{2}} [{u_{2}}^{2}]$

Step 4.3.5.1.2

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

$2 u_{2}$

Step 4.3.5.2

Rewrite the problem using $u_{3}$ and $d u_{3}$ .

$y + \ln (| y |) + C_{3} = - \int {(\sqrt{u_{3}} - 1)}^{2} {\sqrt{u_{3}}}^{- 3} \frac{1}{2} d u_{3}$

Step 4.3.6

Simplify.

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Step 4.3.6.1

Combine $\frac{1}{2}$ and ${(\sqrt{u_{3}} - 1)}^{2}$ .

$y + \ln (| y |) + C_{3} = - \int \frac{{(\sqrt{u_{3}} - 1)}^{2}}{2} {\sqrt{u_{3}}}^{- 3} d u_{3}$

Step 4.3.6.2

Combine $\frac{{(\sqrt{u_{3}} - 1)}^{2}}{2}$ and ${\sqrt{u_{3}}}^{- 3}$ .

$y + \ln (| y |) + C_{3} = - \int \frac{{(\sqrt{u_{3}} - 1)}^{2} {\sqrt{u_{3}}}^{- 3}}{2} d u_{3}$

Step 4.3.6.3

Move ${\sqrt{u_{3}}}^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$y + \ln (| y |) + C_{3} = - \int \frac{{(\sqrt{u_{3}} - 1)}^{2}}{2 {\sqrt{u_{3}}}^{3}} d u_{3}$

Step 4.3.6.4

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

$y + \ln (| y |) + C_{3} = - \int \frac{{(\sqrt{u_{3}} - 1)}^{2}}{2 \sqrt{{u_{3}}^{3}}} d u_{3}$

Step 4.3.7

Since $\frac{1}{2}$ is constant with respect to $u_{3}$ , move $\frac{1}{2}$ out of the integral.

$y + \ln (| y |) + C_{3} = - (\frac{1}{2} \int \frac{{(\sqrt{u_{3}} - 1)}^{2}}{\sqrt{{u_{3}}^{3}}} d u_{3})$

Step 4.3.8

Apply basic rules of exponents.

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Step 4.3.8.1

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

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int \frac{{({u_{3}}^{\frac{1}{2}} - 1)}^{2}}{\sqrt{{u_{3}}^{3}}} d u_{3}$

Step 4.3.8.2

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

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

Step 4.3.8.3

Move ${u_{3}}^{\frac{3}{2}}$ out of the denominator by raising it to the $- 1$ power.

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

Step 4.3.8.4

Multiply the exponents in ${({u_{3}}^{\frac{3}{2}})}^{- 1}$ .

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Step 4.3.8.4.1

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

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int {({u_{3}}^{\frac{1}{2}} - 1)}^{2} {u_{3}}^{\frac{3}{2} \cdot - 1} d u_{3}$

Step 4.3.8.4.2

Multiply $\frac{3}{2} \cdot - 1$ .

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Step 4.3.8.4.2.1

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

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int {({u_{3}}^{\frac{1}{2}} - 1)}^{2} {u_{3}}^{\frac{3 \cdot - 1}{2}} d u_{3}$

Step 4.3.8.4.2.2

Multiply $3$ by $- 1$ .

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

Step 4.3.8.4.3

Move the negative in front of the fraction.

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

Step 4.3.9

Simplify.

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Step 4.3.9.1

Rewrite ${({u_{3}}^{\frac{1}{2}} - 1)}^{2}$ as $({u_{3}}^{\frac{1}{2}} - 1) ({u_{3}}^{\frac{1}{2}} - 1)$ .

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

Step 4.3.9.2

Apply the distributive property.

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

Step 4.3.9.3

Apply the distributive property.

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int ({u_{3}}^{\frac{1}{2}} {u_{3}}^{\frac{1}{2}} + {u_{3}}^{\frac{1}{2}} \cdot - 1 - 1 ({u_{3}}^{\frac{1}{2}} - 1)) {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 4.3.9.4

Apply the distributive property.

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int ({u_{3}}^{\frac{1}{2}} {u_{3}}^{\frac{1}{2}} + {u_{3}}^{\frac{1}{2}} \cdot - 1 - 1 {u_{3}}^{\frac{1}{2}} - 1 \cdot - 1) {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 4.3.9.5

Apply the distributive property.

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int ({u_{3}}^{\frac{1}{2}} {u_{3}}^{\frac{1}{2}} + {u_{3}}^{\frac{1}{2}} \cdot - 1) {u_{3}}^{- \frac{3}{2}} + (- 1 {u_{3}}^{\frac{1}{2}} - 1 \cdot - 1) {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 4.3.9.6

Apply the distributive property.

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int {u_{3}}^{\frac{1}{2}} {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}} + {u_{3}}^{\frac{1}{2}} \cdot - 1 {u_{3}}^{- \frac{3}{2}} + (- 1 {u_{3}}^{\frac{1}{2}} - 1 \cdot - 1) {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 4.3.9.7

Apply the distributive property.

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int {u_{3}}^{\frac{1}{2}} {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}} + {u_{3}}^{\frac{1}{2}} \cdot - 1 {u_{3}}^{- \frac{3}{2}} - 1 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}} - 1 \cdot - 1 {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 4.3.9.8

Reorder ${u_{3}}^{\frac{1}{2}}$ and $- 1$ .

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int {u_{3}}^{\frac{1}{2}} {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}} - 1 \cdot {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}} - 1 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}} - 1 \cdot - 1 {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 4.3.9.9

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

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int {u_{3}}^{\frac{1}{2} + \frac{1}{2}} {u_{3}}^{- \frac{3}{2}} - 1 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}} - 1 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}} - 1 \cdot - 1 {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 4.3.9.10

Combine the numerators over the common denominator.

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int {u_{3}}^{\frac{1 + 1}{2}} {u_{3}}^{- \frac{3}{2}} - 1 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}} - 1 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}} - 1 \cdot - 1 {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 4.3.9.11

Add $1$ and $1$ .

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int {u_{3}}^{\frac{2}{2}} {u_{3}}^{- \frac{3}{2}} - 1 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}} - 1 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}} - 1 \cdot - 1 {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 4.3.9.12

Cancel the common factor of $2$ .

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Step 4.3.9.12.1

Cancel the common factor.

Step 4.3.9.12.2

Rewrite the expression.

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int {u_{3}}^{1} {u_{3}}^{- \frac{3}{2}} - 1 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}} - 1 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}} - 1 \cdot - 1 {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 4.3.9.13

Simplify.

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int u_{3} \cdot {u_{3}}^{- \frac{3}{2}} - 1 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}} - 1 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}} - 1 \cdot - 1 {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 4.3.9.14

Raise $u_{3}$ to the power of $1$ .

Step 4.3.9.15

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

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int {u_{3}}^{1 - \frac{3}{2}} - 1 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}} - 1 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}} - 1 \cdot - 1 {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 4.3.9.16

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

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int {u_{3}}^{\frac{2}{2} - \frac{3}{2}} - 1 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}} - 1 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}} - 1 \cdot - 1 {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 4.3.9.17

Combine the numerators over the common denominator.

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int {u_{3}}^{\frac{2 - 3}{2}} - 1 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}} - 1 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}} - 1 \cdot - 1 {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 4.3.9.18

Subtract $3$ from $2$ .

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int {u_{3}}^{\frac{- 1}{2}} - 1 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}} - 1 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}} - 1 \cdot - 1 {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 4.3.9.19

Factor out negative.

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int {u_{3}}^{\frac{- 1}{2}} - ({u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}}) - 1 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}} - 1 \cdot - 1 {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 4.3.9.20

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

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int {u_{3}}^{\frac{- 1}{2}} - {u_{3}}^{\frac{1}{2} - \frac{3}{2}} - 1 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}} - 1 \cdot - 1 {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 4.3.9.21

Combine the numerators over the common denominator.

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int {u_{3}}^{\frac{- 1}{2}} - {u_{3}}^{\frac{1 - 3}{2}} - 1 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}} - 1 \cdot - 1 {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 4.3.9.22

Subtract $3$ from $1$ .

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int {u_{3}}^{\frac{- 1}{2}} - {u_{3}}^{\frac{- 2}{2}} - 1 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}} - 1 \cdot - 1 {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 4.3.9.23

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

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Step 4.3.9.23.1

Factor $2$ out of $- 2$ .

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int {u_{3}}^{\frac{- 1}{2}} - {u_{3}}^{\frac{2 \cdot - 1}{2}} - 1 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}} - 1 \cdot - 1 {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 4.3.9.23.2

Cancel the common factors.

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Step 4.3.9.23.2.1

Factor $2$ out of $2$ .

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int {u_{3}}^{\frac{- 1}{2}} - {u_{3}}^{\frac{2 \cdot - 1}{2 (1)}} - 1 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}} - 1 \cdot - 1 {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 4.3.9.23.2.2

Cancel the common factor.

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int {u_{3}}^{\frac{- 1}{2}} - {u_{3}}^{\frac{2 \cdot - 1}{2 \cdot 1}} - 1 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}} - 1 \cdot - 1 {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 4.3.9.23.2.3

Rewrite the expression.

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int {u_{3}}^{\frac{- 1}{2}} - {u_{3}}^{\frac{- 1}{1}} - 1 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}} - 1 \cdot - 1 {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 4.3.9.23.2.4

Divide $- 1$ by $1$ .

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int {u_{3}}^{\frac{- 1}{2}} - {u_{3}}^{- 1} - 1 {u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}} - 1 \cdot - 1 {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 4.3.9.24

Factor out negative.

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int {u_{3}}^{\frac{- 1}{2}} - {u_{3}}^{- 1} - ({u_{3}}^{\frac{1}{2}} {u_{3}}^{- \frac{3}{2}}) - 1 \cdot - 1 {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 4.3.9.25

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

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int {u_{3}}^{\frac{- 1}{2}} - {u_{3}}^{- 1} - {u_{3}}^{\frac{1}{2} - \frac{3}{2}} - 1 \cdot - 1 {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 4.3.9.26

Combine the numerators over the common denominator.

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int {u_{3}}^{\frac{- 1}{2}} - {u_{3}}^{- 1} - {u_{3}}^{\frac{1 - 3}{2}} - 1 \cdot - 1 {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 4.3.9.27

Subtract $3$ from $1$ .

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int {u_{3}}^{\frac{- 1}{2}} - {u_{3}}^{- 1} - {u_{3}}^{\frac{- 2}{2}} - 1 \cdot - 1 {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 4.3.9.28

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

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Step 4.3.9.28.1

Factor $2$ out of $- 2$ .

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int {u_{3}}^{\frac{- 1}{2}} - {u_{3}}^{- 1} - {u_{3}}^{\frac{2 \cdot - 1}{2}} - 1 \cdot - 1 {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 4.3.9.28.2

Cancel the common factors.

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Step 4.3.9.28.2.1

Factor $2$ out of $2$ .

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int {u_{3}}^{\frac{- 1}{2}} - {u_{3}}^{- 1} - {u_{3}}^{\frac{2 \cdot - 1}{2 (1)}} - 1 \cdot - 1 {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 4.3.9.28.2.2

Cancel the common factor.

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int {u_{3}}^{\frac{- 1}{2}} - {u_{3}}^{- 1} - {u_{3}}^{\frac{2 \cdot - 1}{2 \cdot 1}} - 1 \cdot - 1 {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 4.3.9.28.2.3

Rewrite the expression.

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int {u_{3}}^{\frac{- 1}{2}} - {u_{3}}^{- 1} - {u_{3}}^{\frac{- 1}{1}} - 1 \cdot - 1 {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 4.3.9.28.2.4

Divide $- 1$ by $1$ .

$y + \ln (| y |) + C_{3} = - \frac{1}{2} \int {u_{3}}^{\frac{- 1}{2}} - {u_{3}}^{- 1} - {u_{3}}^{- 1} - 1 \cdot - 1 {u_{3}}^{- \frac{3}{2}} d u_{3}$

Step 4.3.9.29

Multiply $- 1$ by $- 1$ .

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

Step 4.3.9.30

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

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

Step 4.3.9.31

Subtract ${u_{3}}^{- 1}$ from $- {u_{3}}^{- 1}$ .

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

Step 4.3.9.32

Reorder ${u_{3}}^{\frac{- 1}{2}}$ and $- 2 {u_{3}}^{- 1}$ .

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

Step 4.3.10

Move the negative in front of the fraction.

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

Step 4.3.11

Split the single integral into multiple integrals.

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

Step 4.3.12

Since $- 2$ is constant with respect to $u_{3}$ , move $- 2$ out of the integral.

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

Step 4.3.13

The integral of ${u_{3}}^{- 1}$ with respect to $u_{3}$ is $\ln (| u_{3} |)$ .

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

Step 4.3.14

By the Power Rule, the integral of ${u_{3}}^{- \frac{1}{2}}$ with respect to $u_{3}$ is $2 {u_{3}}^{\frac{1}{2}}$ .

$y + \ln (| y |) + C_{3} = - \frac{1}{2} (- 2 (\ln (| u_{3} |) + C_{4}) + 2 {u_{3}}^{\frac{1}{2}} + C_{5} + \int {u_{3}}^{- \frac{3}{2}} d u_{3})$

Step 4.3.15

By the Power Rule, the integral of ${u_{3}}^{- \frac{3}{2}}$ with respect to $u_{3}$ is $- 2 {u_{3}}^{- \frac{1}{2}}$ .

$y + \ln (| y |) + C_{3} = - \frac{1}{2} (- 2 (\ln (| u_{3} |) + C_{4}) + 2 {u_{3}}^{\frac{1}{2}} + C_{5} - 2 {u_{3}}^{- \frac{1}{2}} + C_{6})$

Step 4.3.16

Simplify.

$y + \ln (| y |) + C_{3} = - \frac{1}{2} (- 2 \ln (| u_{3} |) + 2 {u_{3}}^{\frac{1}{2}} - \frac{2}{{u_{3}}^{\frac{1}{2}}}) + C_{7}$

Step 4.3.17

Substitute back in for each integration substitution variable.

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Step 4.3.17.1

Replace all occurrences of $u_{3}$ with ${u_{2}}^{2}$ .

$y + \ln (| y |) + C_{3} = - \frac{1}{2} (- 2 \ln (| {u_{2}}^{2} |) + 2 {({u_{2}}^{2})}^{\frac{1}{2}} - \frac{2}{{({u_{2}}^{2})}^{\frac{1}{2}}}) + C_{7}$

Step 4.3.17.2

Replace all occurrences of $u_{2}$ with $u_{1} + 1$ .

$y + \ln (| y |) + C_{3} = - \frac{1}{2} (- 2 \ln (| {(u_{1} + 1)}^{2} |) + 2 {({(u_{1} + 1)}^{2})}^{\frac{1}{2}} - \frac{2}{{({(u_{1} + 1)}^{2})}^{\frac{1}{2}}}) + C_{7}$

Step 4.3.17.3

Replace all occurrences of $u_{1}$ with $x - 1$ .

$y + \ln (| y |) + C_{3} = - \frac{1}{2} (- 2 \ln (| {(x - 1 + 1)}^{2} |) + 2 {({(x - 1 + 1)}^{2})}^{\frac{1}{2}} - \frac{2}{{({(x - 1 + 1)}^{2})}^{\frac{1}{2}}}) + C_{7}$

Step 4.3.18

Simplify.

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Step 4.3.18.1

Combine the opposite terms in $- 2 \ln (| {(x - 1 + 1)}^{2} |) + 2 {({(x - 1 + 1)}^{2})}^{\frac{1}{2}} - \frac{2}{{({(x - 1 + 1)}^{2})}^{\frac{1}{2}}}$ .

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Step 4.3.18.1.1

Add $- 1$ and $1$ .

$y + \ln (| y |) + C_{3} = - \frac{1}{2} (- 2 \ln (| {(x + 0)}^{2} |) + 2 {({(x - 1 + 1)}^{2})}^{\frac{1}{2}} - \frac{2}{{({(x - 1 + 1)}^{2})}^{\frac{1}{2}}}) + C_{7}$

Step 4.3.18.1.2

Add $x$ and $0$ .

$y + \ln (| y |) + C_{3} = - \frac{1}{2} (- 2 \ln (| x^{2} |) + 2 {({(x - 1 + 1)}^{2})}^{\frac{1}{2}} - \frac{2}{{({(x - 1 + 1)}^{2})}^{\frac{1}{2}}}) + C_{7}$

Step 4.3.18.1.3

Add $- 1$ and $1$ .

$y + \ln (| y |) + C_{3} = - \frac{1}{2} (- 2 \ln (| x^{2} |) + 2 {({(x + 0)}^{2})}^{\frac{1}{2}} - \frac{2}{{({(x - 1 + 1)}^{2})}^{\frac{1}{2}}}) + C_{7}$

Step 4.3.18.1.4

Add $x$ and $0$ .

$y + \ln (| y |) + C_{3} = - \frac{1}{2} (- 2 \ln (| x^{2} |) + 2 {(x^{2})}^{\frac{1}{2}} - \frac{2}{{({(x - 1 + 1)}^{2})}^{\frac{1}{2}}}) + C_{7}$

Step 4.3.18.1.5

Add $- 1$ and $1$ .

$y + \ln (| y |) + C_{3} = - \frac{1}{2} (- 2 \ln (| x^{2} |) + 2 {(x^{2})}^{\frac{1}{2}} - \frac{2}{{({(x + 0)}^{2})}^{\frac{1}{2}}}) + C_{7}$

Step 4.3.18.1.6

Add $x$ and $0$ .

$y + \ln (| y |) + C_{3} = - \frac{1}{2} (- 2 \ln (| x^{2} |) + 2 {(x^{2})}^{\frac{1}{2}} - \frac{2}{{(x^{2})}^{\frac{1}{2}}}) + C_{7}$

Step 4.3.18.2

Simplify each term.

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Step 4.3.18.2.1

Remove non-negative terms from the absolute value.

$y + \ln (| y |) + C_{3} = - \frac{1}{2} (- 2 \ln (x^{2}) + 2 {(x^{2})}^{\frac{1}{2}} - \frac{2}{{(x^{2})}^{\frac{1}{2}}}) + C_{7}$

Step 4.3.18.2.2

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

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Step 4.3.18.2.2.1

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

$y + \ln (| y |) + C_{3} = - \frac{1}{2} (- 2 \ln (x^{2}) + 2 x^{2 (\frac{1}{2})} - \frac{2}{{(x^{2})}^{\frac{1}{2}}}) + C_{7}$

Step 4.3.18.2.2.2

Cancel the common factor of $2$ .

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Step 4.3.18.2.2.2.1

Cancel the common factor.

$y + \ln (| y |) + C_{3} = - \frac{1}{2} (- 2 \ln (x^{2}) + 2 x^{2 (\frac{1}{2})} - \frac{2}{{(x^{2})}^{\frac{1}{2}}}) + C_{7}$

Step 4.3.18.2.2.2.2

Rewrite the expression.

$y + \ln (| y |) + C_{3} = - \frac{1}{2} (- 2 \ln (x^{2}) + 2 x^{1} - \frac{2}{{(x^{2})}^{\frac{1}{2}}}) + C_{7}$

Step 4.3.18.2.3

Simplify.

$y + \ln (| y |) + C_{3} = - \frac{1}{2} (- 2 \ln (x^{2}) + 2 x - \frac{2}{{(x^{2})}^{\frac{1}{2}}}) + C_{7}$

Step 4.3.18.2.4

Simplify the denominator.

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Step 4.3.18.2.4.1

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

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Step 4.3.18.2.4.1.1

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

$y + \ln (| y |) + C_{3} = - \frac{1}{2} (- 2 \ln (x^{2}) + 2 x - \frac{2}{x^{2 (\frac{1}{2})}}) + C_{7}$

Step 4.3.18.2.4.1.2

Cancel the common factor of $2$ .

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Step 4.3.18.2.4.1.2.1

Cancel the common factor.

$y + \ln (| y |) + C_{3} = - \frac{1}{2} (- 2 \ln (x^{2}) + 2 x - \frac{2}{x^{2 (\frac{1}{2})}}) + C_{7}$

Step 4.3.18.2.4.1.2.2

Rewrite the expression.

$y + \ln (| y |) + C_{3} = - \frac{1}{2} (- 2 \ln (x^{2}) + 2 x - \frac{2}{x^{1}}) + C_{7}$

Step 4.3.18.2.4.2

Simplify.

$y + \ln (| y |) + C_{3} = - \frac{1}{2} (- 2 \ln (x^{2}) + 2 x - \frac{2}{x}) + C_{7}$

Step 4.3.18.3

Apply the distributive property.

$y + \ln (| y |) + C_{3} = - \frac{1}{2} (- 2 \ln (x^{2})) - \frac{1}{2} (2 x) - \frac{1}{2} (- \frac{2}{x}) + C_{7}$

Step 4.3.18.4

Simplify.

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Step 4.3.18.4.1

Cancel the common factor of $2$ .

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Step 4.3.18.4.1.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$y + \ln (| y |) + C_{3} = \frac{- 1}{2} (- 2 \ln (x^{2})) - \frac{1}{2} (2 x) - \frac{1}{2} (- \frac{2}{x}) + C_{7}$

Step 4.3.18.4.1.2

Factor $2$ out of $- 2 \ln (x^{2})$ .

$y + \ln (| y |) + C_{3} = \frac{- 1}{2} (2 (- \ln (x^{2}))) - \frac{1}{2} (2 x) - \frac{1}{2} (- \frac{2}{x}) + C_{7}$

Step 4.3.18.4.1.3

Cancel the common factor.

$y + \ln (| y |) + C_{3} = \frac{- 1}{2} (2 (- \ln (x^{2}))) - \frac{1}{2} (2 x) - \frac{1}{2} (- \frac{2}{x}) + C_{7}$

Step 4.3.18.4.1.4

Rewrite the expression.

$y + \ln (| y |) + C_{3} = - - \ln (x^{2}) - \frac{1}{2} (2 x) - \frac{1}{2} (- \frac{2}{x}) + C_{7}$

Step 4.3.18.4.2

Multiply $- 1$ by $- 1$ .

$y + \ln (| y |) + C_{3} = 1 \ln (x^{2}) - \frac{1}{2} (2 x) - \frac{1}{2} (- \frac{2}{x}) + C_{7}$

Step 4.3.18.4.3

Multiply $\ln (x^{2})$ by $1$ .

$y + \ln (| y |) + C_{3} = \ln (x^{2}) - \frac{1}{2} (2 x) - \frac{1}{2} (- \frac{2}{x}) + C_{7}$

Step 4.3.18.4.4

Cancel the common factor of $2$ .

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Step 4.3.18.4.4.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$y + \ln (| y |) + C_{3} = \ln (x^{2}) + \frac{- 1}{2} (2 x) - \frac{1}{2} (- \frac{2}{x}) + C_{7}$

Step 4.3.18.4.4.2

Factor $2$ out of $2 x$ .

$y + \ln (| y |) + C_{3} = \ln (x^{2}) + \frac{- 1}{2} (2 (x)) - \frac{1}{2} (- \frac{2}{x}) + C_{7}$

Step 4.3.18.4.4.3

Cancel the common factor.

$y + \ln (| y |) + C_{3} = \ln (x^{2}) + \frac{- 1}{2} (2 x) - \frac{1}{2} (- \frac{2}{x}) + C_{7}$

Step 4.3.18.4.4.4

Rewrite the expression.

$y + \ln (| y |) + C_{3} = \ln (x^{2}) - x - \frac{1}{2} (- \frac{2}{x}) + C_{7}$

Step 4.3.18.4.5

Cancel the common factor of $2$ .

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Step 4.3.18.4.5.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$y + \ln (| y |) + C_{3} = \ln (x^{2}) - x + \frac{- 1}{2} (- \frac{2}{x}) + C_{7}$

Step 4.3.18.4.5.2

Move the leading negative in $- \frac{2}{x}$ into the numerator.

$y + \ln (| y |) + C_{3} = \ln (x^{2}) - x + \frac{- 1}{2} \cdot \frac{- 2}{x} + C_{7}$

Step 4.3.18.4.5.3

Factor $2$ out of $- 2$ .

$y + \ln (| y |) + C_{3} = \ln (x^{2}) - x + \frac{- 1}{2} \cdot \frac{2 (- 1)}{x} + C_{7}$

Step 4.3.18.4.5.4

Cancel the common factor.

$y + \ln (| y |) + C_{3} = \ln (x^{2}) - x + \frac{- 1}{2} \cdot \frac{2 \cdot - 1}{x} + C_{7}$

Step 4.3.18.4.5.5

Rewrite the expression.

$y + \ln (| y |) + C_{3} = \ln (x^{2}) - x - \frac{- 1}{x} + C_{7}$

Step 4.3.18.5

Simplify each term.

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Step 4.3.18.5.1

Move the negative in front of the fraction.

$y + \ln (| y |) + C_{3} = \ln (x^{2}) - x - - \frac{1}{x} + C_{7}$

Step 4.3.18.5.2

Multiply $- - \frac{1}{x}$ .

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Step 4.3.18.5.2.1

Multiply $- 1$ by $- 1$ .

$y + \ln (| y |) + C_{3} = \ln (x^{2}) - x + 1 \frac{1}{x} + C_{7}$

Step 4.3.18.5.2.2

Multiply $\frac{1}{x}$ by $1$ .

$y + \ln (| y |) + C_{3} = \ln (x^{2}) - x + \frac{1}{x} + C_{7}$

Step 4.3.19

Reorder terms.

$y + \ln (| y |) + C_{3} = - x + \frac{1}{x} + \ln (x^{2}) + C_{7}$

Step 4.4

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

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