Calculus Examples

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

Step 1

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

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

Step 2

Multiply both sides by $\frac{1}{x - 1}$ .

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

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $x - 1$ .

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Step 3.1.1

Factor $x - 1$ out of $y (x - 1)$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

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

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

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

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

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.

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

Step 4.3.2

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

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Step 4.3.2.1

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 4.3.2.2

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 4.3.2.3

Multiply the new quotient term by the divisor.

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

Step 4.3.2.4

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 4.3.2.5

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 4.3.2.6

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 4.3.2.7

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 4.3.2.8

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 4.3.2.9

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 4.3.2.10

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 4.3.2.11

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

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

Step 4.3.3

Split the single integral into multiple integrals.

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

Step 4.3.4

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

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

Step 4.3.5

Apply the constant rule.

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

Step 4.3.6

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

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

Step 4.3.7

Let $u = x - 1$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.3.7.1

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

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Step 4.3.7.1.1

Differentiate $x - 1$ .

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

Step 4.3.7.1.2

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 4.3.7.1.3

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 4.3.7.1.4

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

$1 + 0$

Step 4.3.7.1.5

Add $1$ and $0$ .

$1$

Step 4.3.7.2

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

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

Step 4.3.8

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

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

Step 4.3.9

Simplify.

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

Step 4.3.10

Replace all occurrences of $u$ with $x - 1$ .

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

Step 4.3.11

Simplify.

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Step 4.3.11.1

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

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

Step 4.3.11.2

To write $\ln (| x - 1 |)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 4.3.11.3

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

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

Step 4.3.11.4

Combine the numerators over the common denominator.

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

Step 4.3.11.5

Move $2$ to the left of $\ln (| x - 1 |)$ .

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

Step 4.3.11.6

Apply the distributive property.

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

Step 4.3.12

Reorder terms.

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

Step 4.3.13

Reorder terms.

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

Step 4.4

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

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

Step 5

Solve for $y$ .

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Step 5.1

Multiply both sides of the equation by $2$ .

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

Step 5.2

Simplify both sides of the equation.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

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

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Step 5.2.1.1.1

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

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

Step 5.2.1.1.2

Cancel the common factor of $2$ .

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Step 5.2.1.1.2.1

Cancel the common factor.

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

Step 5.2.1.1.2.2

Rewrite the expression.

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

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

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

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Step 5.2.2.1.1

Simplify each term.

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Step 5.2.2.1.1.1

Apply the distributive property.

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

Step 5.2.2.1.1.2

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

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

Step 5.2.2.1.1.3

Cancel the common factor of $2$ .

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Step 5.2.2.1.1.3.1

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

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

Step 5.2.2.1.1.3.2

Factor $2$ out of $2 \ln (| x - 1 |)$ .

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

Step 5.2.2.1.1.3.3

Cancel the common factor.

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

Step 5.2.2.1.1.3.4

Rewrite the expression.

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

Step 5.2.2.1.1.4

Rewrite $- 1 \ln (| x - 1 |)$ as $- \ln (| x - 1 |)$ .

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

Step 5.2.2.1.1.5

To write $- \ln (| x - 1 |)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 5.2.2.1.1.6

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

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

Step 5.2.2.1.1.7

Combine the numerators over the common denominator.

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

Step 5.2.2.1.1.8

Multiply $2$ by $- 1$ .

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

Step 5.2.2.1.2

Apply the distributive property.

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

Step 5.2.2.1.3

Simplify.

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Step 5.2.2.1.3.1

Cancel the common factor of $2$ .

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Step 5.2.2.1.3.1.1

Cancel the common factor.

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

Step 5.2.2.1.3.1.2

Rewrite the expression.

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

Step 5.2.2.1.3.2

Multiply $- 1$ by $2$ .

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

Step 5.3

Simplify $- 2 \ln (| x - 1 |)$ by moving $2$ inside the logarithm.

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

Step 5.4

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

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

Step 5.5

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

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Step 5.5.1

Remove the absolute value in ${| x - 1 |}^{2}$ because exponentiations with even powers are always positive.

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

Step 5.5.2

Rewrite $- x^{2} - \ln ({(x - 1)}^{2}) - 2 x + 2 C$ in a factored form.

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Step 5.5.2.1

Regroup terms.

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

Step 5.5.2.2

Factor $- 1$ out of $- x^{2} - 2 x + 2 C$ .

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Step 5.5.2.2.1

Move $- 2 x$ .

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

Step 5.5.2.2.2

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

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

Step 5.5.2.2.3

Factor $- 1$ out of $2 C$ .

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

Step 5.5.2.2.4

Factor $- 1$ out of $- 2 x$ .

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

Step 5.5.2.2.5

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

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

Step 5.5.2.2.6

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

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

Step 5.5.2.3

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

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Step 5.5.2.3.1

Reorder $- \ln ({(x - 1)}^{2})$ and $- (x^{2} - 2 C + 2 x)$ .

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

Step 5.5.2.3.2

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

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

Step 5.5.2.3.3

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

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

Step 5.6

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

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Step 5.6.1

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

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

Step 5.6.2

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

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

Step 5.6.3

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

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

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

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

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

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

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

Step 6

Simplify the constant of integration.

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

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