Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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

Step 1.2

Simplify.

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Step 1.2.1

Combine.

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

Step 1.2.2

Cancel the common factor of ${(y - 1)}^{2}$ .

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Step 1.2.2.1

Cancel the common factor.

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

Step 1.2.2.2

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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Step 2.2.1.1.1

Differentiate $y - 1$ .

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

Step 2.2.1.1.2

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 2.2.1.1.3

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 2.2.1.1.4

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

$1 + 0$

Step 2.2.1.1.5

Add $1$ and $0$ .

$1$

Step 2.2.1.2

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

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

Step 2.2.2

Apply basic rules of exponents.

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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Step 2.2.2.2.1

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

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

Step 2.2.2.2.2

Multiply $2$ by $- 1$ .

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

Step 2.2.3

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

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

Step 2.2.4

Rewrite $- u^{- 1} + C_{1}$ as $- \frac{1}{u} + C_{1}$ .

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

Step 2.2.5

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Find the LCD of the terms in the equation.

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Step 3.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y - 1, 1, 1$

Step 3.1.2

Remove parentheses.

$y - 1, 1, 1$

Step 3.1.3

The LCM of one and any expression is the expression.

$y - 1$

Step 3.2

Multiply each term in $- \frac{1}{y - 1} = \ln (| x |) + C$ by $y - 1$ to eliminate the fractions.

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Step 3.2.1

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

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $y - 1$ .

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Step 3.2.2.1.1

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

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

Step 3.2.2.1.2

Cancel the common factor.

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

Step 3.2.2.1.3

Rewrite the expression.

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Apply the distributive property.

$- 1 = \ln (| x |) y + \ln (| x |) \cdot - 1 + C (y - 1)$

Step 3.2.3.1.2

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

$- 1 = \ln (| x |) y - 1 \cdot \ln (| x |) + C (y - 1)$

Step 3.2.3.1.3

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

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

Step 3.2.3.1.4

Apply the distributive property.

$- 1 = \ln (| x |) y - \ln (| x |) + C y + C \cdot - 1$

Step 3.2.3.1.5

Move $- 1$ to the left of $C$ .

$- 1 = \ln (| x |) y - \ln (| x |) + C y - 1 \cdot C$

Step 3.2.3.1.6

Rewrite $- 1 C$ as $- C$ .

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

Step 3.2.3.2

Reorder factors in $\ln (| x |) y - \ln (| x |) + C y - C$ .

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

Step 3.3

Solve the equation.

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Step 3.3.1

Rewrite the equation as $y \ln (| x |) - \ln (| x |) + C y - C = - 1$ .

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

Step 3.3.2

Move all the terms containing a logarithm to the left side of the equation.

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

Step 3.3.3

Add $C y$ to both sides of the equation.

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

Step 3.3.4

Add $\ln (| x |)$ to both sides of the equation.

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

Step 3.3.5

Factor $y$ out of $y \ln (| x |) + C y$ .

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Step 3.3.5.1

Factor $y$ out of $C y$ .

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

Step 3.3.5.2

Factor $y$ out of $y \ln (| x |) + y C$ .

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

Step 3.3.6

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

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Step 3.3.6.1

Divide each term in $y (\ln (| x |) + C) = C - 1 + \ln (| x |)$ by $\ln (| x |) + C$ .

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

Step 3.3.6.2

Simplify the left side.

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Step 3.3.6.2.1

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

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Step 3.3.6.2.1.1

Cancel the common factor.

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

Step 3.3.6.2.1.2

Divide $y$ by $1$ .

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

Step 3.3.6.3

Simplify the right side.

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Step 3.3.6.3.1

Move the negative in front of the fraction.

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

Step 3.3.6.3.2

Combine the numerators over the common denominator.

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

Step 3.3.6.3.3

Combine the numerators over the common denominator.

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

Step 4

Simplify the constant of integration.

$y = \frac{C + \ln (| x |)}{\ln (| x |) + C}$