Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Solve for $\frac{d y}{d x}$ .

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Step 1.1.1

Apply the distributive property.

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

Step 1.1.2

Move all terms not containing $\frac{d y}{d x}$ to the right side of the equation.

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Step 1.1.2.1

Subtract $x^{2}$ from both sides of the equation.

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

Step 1.1.2.2

Add $x^{2} y$ to both sides of the equation.

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

Step 1.1.3

Factor $y^{2} \frac{d y}{d x}$ out of $y^{2} \frac{d y}{d x} + x y^{2} \frac{d y}{d x}$ .

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Step 1.1.3.1

Factor $y^{2} \frac{d y}{d x}$ out of $y^{2} \frac{d y}{d x}$ .

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

Step 1.1.3.2

Factor $y^{2} \frac{d y}{d x}$ out of $x y^{2} \frac{d y}{d x}$ .

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

Step 1.1.3.3

Factor $y^{2} \frac{d y}{d x}$ out of $y^{2} \frac{d y}{d x} \cdot 1 + y^{2} \frac{d y}{d x} x$ .

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

Step 1.1.4

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

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Step 1.1.4.1

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

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

Step 1.1.4.2

Simplify the left side.

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Step 1.1.4.2.1

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

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Step 1.1.4.2.1.1

Cancel the common factor.

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

Step 1.1.4.2.1.2

Rewrite the expression.

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

Step 1.1.4.2.2

Cancel the common factor of $1 + x$ .

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Step 1.1.4.2.2.1

Cancel the common factor.

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

Step 1.1.4.2.2.2

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

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

Step 1.1.4.3

Simplify the right side.

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Step 1.1.4.3.1

Simplify each term.

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Step 1.1.4.3.1.1

Move the negative in front of the fraction.

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

Step 1.1.4.3.1.2

Cancel the common factor of $y$ and $y^{2}$ .

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Step 1.1.4.3.1.2.1

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

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

Step 1.1.4.3.1.2.2

Cancel the common factors.

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Step 1.1.4.3.1.2.2.1

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

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

Step 1.1.4.3.1.2.2.2

Cancel the common factor.

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

Step 1.1.4.3.1.2.2.3

Rewrite the expression.

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

Step 1.2

Factor.

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Step 1.2.1

To write $\frac{x^{2}}{y (1 + x)}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

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

Step 1.2.2

Write each expression with a common denominator of $(1 + x) y^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.2.1

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

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

Step 1.2.2.2

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

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

Step 1.2.2.3

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

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

Step 1.2.2.4

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

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

Step 1.2.2.5

Add $1$ and $1$ .

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

Step 1.2.3

Combine the numerators over the common denominator.

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

Step 1.2.4

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

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Step 1.2.4.1

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

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

Step 1.2.4.2

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

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

Step 1.2.4.3

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

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

Step 1.2.4.4

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

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

Step 1.3

Regroup factors.

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

Step 1.4

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

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

Step 1.5

Simplify.

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Step 1.5.1

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

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

Step 1.5.2

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

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Step 1.5.2.1

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

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

Step 1.5.2.2

Cancel the common factor.

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

Step 1.5.2.3

Rewrite the expression.

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

Step 1.5.3

Cancel the common factor of $- 1 + y$ .

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Step 1.5.3.1

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

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

Step 1.5.3.2

Cancel the common factor.

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

Step 1.5.3.3

Rewrite the expression.

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

Step 1.6

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Reorder $- 1$ and $y$ .

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

Step 2.2.2

Divide $y^{2}$ by $y - 1$ .

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Step 2.2.2.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$y$

$1$

$y^{2}$

$0 y$

$0$

Step 2.2.2.2

Divide the highest order term in the dividend $y^{2}$ by the highest order term in divisor $y$ .

					$y$
	$y$	-	$1$		$y^{2}$	+	$0 y$	+	$0$

Step 2.2.2.3

Multiply the new quotient term by the divisor.

				$y$
$y$	-	$1$		$y^{2}$	+	$0 y$	+	$0$
			+	$y^{2}$	-	$y$

Step 2.2.2.4

The expression needs to be subtracted from the dividend, so change all the signs in $y^{2} - y$

				$y$
$y$	-	$1$		$y^{2}$	+	$0 y$	+	$0$
			-	$y^{2}$	+	$y$

Step 2.2.2.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$y$
$y$	-	$1$		$y^{2}$	+	$0 y$	+	$0$
			-	$y^{2}$	+	$y$
					+	$y$

Step 2.2.2.6

Pull the next terms from the original dividend down into the current dividend.

				$y$
$y$	-	$1$		$y^{2}$	+	$0 y$	+	$0$
			-	$y^{2}$	+	$y$
					+	$y$	+	$0$

Step 2.2.2.7

Divide the highest order term in the dividend $y$ by the highest order term in divisor $y$ .

				$y$	+	$1$
$y$	-	$1$		$y^{2}$	+	$0 y$	+	$0$
			-	$y^{2}$	+	$y$
					+	$y$	+	$0$

Step 2.2.2.8

Multiply the new quotient term by the divisor.

				$y$	+	$1$
$y$	-	$1$		$y^{2}$	+	$0 y$	+	$0$
			-	$y^{2}$	+	$y$
					+	$y$	+	$0$
					+	$y$	-	$1$

Step 2.2.2.9

The expression needs to be subtracted from the dividend, so change all the signs in $y - 1$

				$y$	+	$1$
$y$	-	$1$		$y^{2}$	+	$0 y$	+	$0$
			-	$y^{2}$	+	$y$
					+	$y$	+	$0$
					-	$y$	+	$1$

Step 2.2.2.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$y$	+	$1$
$y$	-	$1$		$y^{2}$	+	$0 y$	+	$0$
			-	$y^{2}$	+	$y$
					+	$y$	+	$0$
					-	$y$	+	$1$
							+	$1$

Step 2.2.2.11

The final answer is the quotient plus the remainder over the divisor.

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

Step 2.2.3

Split the single integral into multiple integrals.

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

Step 2.2.4

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

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

Step 2.2.5

Apply the constant rule.

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

Step 2.2.6

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

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Step 2.2.6.1

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

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Step 2.2.6.1.1

Differentiate $y - 1$ .

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

Step 2.2.6.1.2

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

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

Step 2.2.6.1.3

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

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

Step 2.2.6.1.4

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

$1 + 0$

Step 2.2.6.1.5

Add $1$ and $0$ .

$1$

Step 2.2.6.2

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

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

Step 2.2.7

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

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

Step 2.2.8

Simplify.

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

Step 2.2.9

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Reorder $1$ and $x$ .

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

Step 2.3.2

Divide $x^{2}$ by $x + 1$ .

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Step 2.3.2.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$1$

$x^{2}$

$0 x$

$0$

Step 2.3.2.2

Divide the highest order term in the dividend $x^{2}$ by the highest order term in divisor $x$ .

					$x$
	$x$	+	$1$		$x^{2}$	+	$0 x$	+	$0$

Step 2.3.2.3

Multiply the new quotient term by the divisor.

				$x$
$x$	+	$1$		$x^{2}$	+	$0 x$	+	$0$
			+	$x^{2}$	+	$x$

Step 2.3.2.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{2} + x$

				$x$
$x$	+	$1$		$x^{2}$	+	$0 x$	+	$0$
			-	$x^{2}$	-	$x$

Step 2.3.2.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x$
$x$	+	$1$		$x^{2}$	+	$0 x$	+	$0$
			-	$x^{2}$	-	$x$
					-	$x$

Step 2.3.2.6

Pull the next terms from the original dividend down into the current dividend.

				$x$
$x$	+	$1$		$x^{2}$	+	$0 x$	+	$0$
			-	$x^{2}$	-	$x$
					-	$x$	+	$0$

Step 2.3.2.7

Divide the highest order term in the dividend $- x$ by the highest order term in divisor $x$ .

				$x$	-	$1$
$x$	+	$1$		$x^{2}$	+	$0 x$	+	$0$
			-	$x^{2}$	-	$x$
					-	$x$	+	$0$

Step 2.3.2.8

Multiply the new quotient term by the divisor.

				$x$	-	$1$
$x$	+	$1$		$x^{2}$	+	$0 x$	+	$0$
			-	$x^{2}$	-	$x$
					-	$x$	+	$0$
					-	$x$	-	$1$

Step 2.3.2.9

The expression needs to be subtracted from the dividend, so change all the signs in $- x - 1$

				$x$	-	$1$
$x$	+	$1$		$x^{2}$	+	$0 x$	+	$0$
			-	$x^{2}$	-	$x$
					-	$x$	+	$0$
					+	$x$	+	$1$

Step 2.3.2.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x$	-	$1$
$x$	+	$1$		$x^{2}$	+	$0 x$	+	$0$
			-	$x^{2}$	-	$x$
					-	$x$	+	$0$
					+	$x$	+	$1$
							+	$1$

Step 2.3.2.11

The final answer is the quotient plus the remainder over the divisor.

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

Step 2.3.3

Split the single integral into multiple integrals.

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

Step 2.3.4

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

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

Step 2.3.5

Apply the constant rule.

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

Step 2.3.6

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

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Step 2.3.6.1

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

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Step 2.3.6.1.1

Differentiate $x + 1$ .

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

Step 2.3.6.1.2

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 2.3.6.1.3

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 2.3.6.1.4

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

$1 + 0$

Step 2.3.6.1.5

Add $1$ and $0$ .

$1$

Step 2.3.6.2

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

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

Step 2.3.7

The integral of $\frac{1}{u_{2}}$ with respect to $u_{2}$ is $\ln (| u_{2} |)$ .

$\frac{1}{2} y^{2} + y + \ln (| y - 1 |) + C_{4} = \frac{1}{2} x^{2} + C_{5} - x + C_{6} + \ln (| u_{2} |) + C_{7}$

Step 2.3.8

Simplify.

$\frac{1}{2} y^{2} + y + \ln (| y - 1 |) + C_{4} = \frac{1}{2} x^{2} - x + \ln (| u_{2} |) + C_{8}$

Step 2.3.9

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

$\frac{1}{2} y^{2} + y + \ln (| y - 1 |) + C_{4} = \frac{1}{2} x^{2} - x + \ln (| x + 1 |) + C_{8}$

Step 2.4

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

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