Calculus Examples

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

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

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

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

Step 1.1.2

Simplify the left side.

Tap for more steps...

Step 1.1.2.1

Cancel the common factor of $y$ .

Tap for more steps...

Step 1.1.2.1.1

Cancel the common factor.

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

Step 1.1.2.1.2

Rewrite the expression.

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

Step 1.1.2.2

Cancel the common factor of $\ln (x)$ .

Tap for more steps...

Step 1.1.2.2.1

Cancel the common factor.

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

Step 1.1.2.2.2

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

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

Step 1.1.3

Simplify the right side.

Tap for more steps...

Step 1.1.3.1

Apply the product rule to $\frac{y - 1}{x}$ .

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

Step 1.1.3.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.1.3.3

Combine.

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

Step 1.1.3.4

Multiply ${(y - 1)}^{2}$ by $1$ .

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

Step 1.2

Regroup factors.

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

Step 1.3

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

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

Step 1.4

Simplify.

Tap for more steps...

Step 1.4.1

Combine.

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

Step 1.4.2

Cancel the common factor of $x^{2} \ln (x)$ .

Tap for more steps...

Step 1.4.2.1

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

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

Step 1.4.2.2

Cancel the common factor.

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

Step 1.4.2.3

Rewrite the expression.

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

Step 1.4.3

Multiply ${(y - 1)}^{2}$ by $1$ .

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

Step 1.5

Rewrite the equation.

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

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

Tap for more steps...

Step 2.2.1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (x)$ and $d v = x^{2}$ .

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

Step 2.2.2

Simplify.

Tap for more steps...

Step 2.2.2.1

Combine $\frac{1}{3}$ and $x^{3}$ .

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

Step 2.2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Simplify.

Tap for more steps...

Step 2.2.4.1

Combine $x^{3}$ and $\frac{1}{x}$ .

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

Step 2.2.4.2

Cancel the common factor of $x^{3}$ and $x$ .

Tap for more steps...

Step 2.2.4.2.1

Factor $x$ out of $x^{3}$ .

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

Step 2.2.4.2.2

Cancel the common factors.

Tap for more steps...

Step 2.2.4.2.2.1

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

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

Step 2.2.4.2.2.2

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

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

Step 2.2.4.2.2.3

Cancel the common factor.

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

Step 2.2.4.2.2.4

Rewrite the expression.

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

Step 2.2.4.2.2.5

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

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

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

Step 2.2.5

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

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

Step 2.2.6

Simplify the answer.

Tap for more steps...

Step 2.2.6.1

Rewrite $\frac{\ln (x) x^{3}}{3} - \frac{1}{3} (\frac{1}{3} x^{3} + C_{1})$ as $\frac{1}{3} \ln (x) x^{3} - \frac{1}{3} \cdot \frac{1}{3} x^{3} + C_{1}$ .

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

Step 2.2.6.2

Simplify.

Tap for more steps...

Step 2.2.6.2.1

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

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

Step 2.2.6.2.2

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

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

Step 2.2.6.2.3

Multiply $\frac{1}{3}$ by $\frac{1}{3}$ .

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

Step 2.2.6.2.4

Multiply $3$ by $3$ .

$\frac{\ln (x) x^{3}}{3} - \frac{1}{9} x^{3} + C_{1} = \int \frac{{(y - 1)}^{2}}{y} d y$

Step 2.2.6.3

Combine $x^{3}$ and $\frac{1}{9}$ .

$\frac{\ln (x) x^{3}}{3} - \frac{x^{3}}{9} + C_{1} = \int \frac{{(y - 1)}^{2}}{y} d y$

Step 2.2.6.4

Reorder terms.

$\frac{1}{3} \ln (x) x^{3} - \frac{1}{9} x^{3} + C_{1} = \int \frac{{(y - 1)}^{2}}{y} d y$

Step 2.2.7

Reorder terms.

$\frac{1}{3} x^{3} \ln (x) - \frac{1}{9} x^{3} + C_{1} = \int \frac{{(y - 1)}^{2}}{y} d y$

Step 2.3

Integrate the right side.

Tap for more steps...

Step 2.3.1

Rewrite ${(y - 1)}^{2}$ as $(y - 1) (y - 1)$ .

$\frac{1}{3} x^{3} \ln (x) - \frac{1}{9} x^{3} + C_{1} = \int \frac{(y - 1) (y - 1)}{y} d y$

Step 2.3.2

Apply the distributive property.

$\frac{1}{3} x^{3} \ln (x) - \frac{1}{9} x^{3} + C_{1} = \int \frac{y (y - 1) - 1 (y - 1)}{y} d y$

Step 2.3.3

Apply the distributive property.

$\frac{1}{3} x^{3} \ln (x) - \frac{1}{9} x^{3} + C_{1} = \int \frac{y \cdot y + y \cdot - 1 - 1 (y - 1)}{y} d y$

Step 2.3.4

Apply the distributive property.

$\frac{1}{3} x^{3} \ln (x) - \frac{1}{9} x^{3} + C_{1} = \int \frac{y \cdot y + y \cdot - 1 - 1 y - 1 \cdot - 1}{y} d y$

Step 2.3.5

Reorder $y$ and $- 1$ .

$\frac{1}{3} x^{3} \ln (x) - \frac{1}{9} x^{3} + C_{1} = \int \frac{y \cdot y - 1 \cdot y - 1 y - 1 \cdot - 1}{y} d y$

Step 2.3.6

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

$\frac{1}{3} x^{3} \ln (x) - \frac{1}{9} x^{3} + C_{1} = \int \frac{y^{1} y - 1 y - 1 y - 1 \cdot - 1}{y} d y$

Step 2.3.7

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

$\frac{1}{3} x^{3} \ln (x) - \frac{1}{9} x^{3} + C_{1} = \int \frac{y^{1} y^{1} - 1 y - 1 y - 1 \cdot - 1}{y} d y$

Step 2.3.8

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

$\frac{1}{3} x^{3} \ln (x) - \frac{1}{9} x^{3} + C_{1} = \int \frac{y^{1 + 1} - 1 y - 1 y - 1 \cdot - 1}{y} d y$

Step 2.3.9

Simplify the expression.

Tap for more steps...

Step 2.3.9.1

Add $1$ and $1$ .

$\frac{1}{3} x^{3} \ln (x) - \frac{1}{9} x^{3} + C_{1} = \int \frac{y^{2} - 1 y - 1 y - 1 \cdot - 1}{y} d y$

Step 2.3.9.2

Multiply $- 1$ by $- 1$ .

$\frac{1}{3} x^{3} \ln (x) - \frac{1}{9} x^{3} + C_{1} = \int \frac{y^{2} - 1 y - 1 y + 1}{y} d y$

Step 2.3.10

Subtract $1 y$ from $- 1 y$ .

$\frac{1}{3} x^{3} \ln (x) - \frac{1}{9} x^{3} + C_{1} = \int \frac{y^{2} - 2 y + 1}{y} d y$

Step 2.3.11

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

Tap for more steps...

Step 2.3.11.1

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

$y$

$0$

$y^{2}$

$2 y$

$1$

Step 2.3.11.2

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

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

Step 2.3.11.3

Multiply the new quotient term by the divisor.

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

Step 2.3.11.4

The expression needs to be subtracted from the dividend, so change all the signs in $y^{2} + 0$

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

Step 2.3.11.5

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

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

Step 2.3.11.6

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

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

Step 2.3.11.7

Divide the highest order term in the dividend $- 2 y$ by the highest order term in divisor $y$ .

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

Step 2.3.11.8

Multiply the new quotient term by the divisor.

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

Step 2.3.11.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 y + 0$

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

Step 2.3.11.10

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

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

Step 2.3.11.11

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

$\frac{1}{3} x^{3} \ln (x) - \frac{1}{9} x^{3} + C_{1} = \int y - 2 + \frac{1}{y} d y$

Step 2.3.12

Split the single integral into multiple integrals.

$\frac{1}{3} x^{3} \ln (x) - \frac{1}{9} x^{3} + C_{1} = \int y d y + \int - 2 d y + \int \frac{1}{y} d y$

Step 2.3.13

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

$\frac{1}{3} x^{3} \ln (x) - \frac{1}{9} x^{3} + C_{1} = \frac{1}{2} y^{2} + C_{2} + \int - 2 d y + \int \frac{1}{y} d y$

Step 2.3.14

Apply the constant rule.

$\frac{1}{3} x^{3} \ln (x) - \frac{1}{9} x^{3} + C_{1} = \frac{1}{2} y^{2} + C_{2} - 2 y + C_{3} + \int \frac{1}{y} d y$

Step 2.3.15

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

$\frac{1}{3} x^{3} \ln (x) - \frac{1}{9} x^{3} + C_{1} = \frac{1}{2} y^{2} + C_{2} - 2 y + C_{3} + \ln (| y |) + C_{4}$

Step 2.3.16

Simplify.

$\frac{1}{3} x^{3} \ln (x) - \frac{1}{9} x^{3} + C_{1} = \frac{1}{2} y^{2} - 2 y + \ln (| y |) + C_{5}$

Step 2.4

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

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