Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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Step 1.1.1

Multiply by $1$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.3

Regroup factors.

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

Step 1.4

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

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

Step 1.5

Simplify.

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Step 1.5.1

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

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

Step 1.5.2

Cancel the common factor of $y - 1$ .

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Step 1.5.2.1

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

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

Step 1.5.2.2

Cancel the common factor.

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

Step 1.5.2.3

Rewrite the expression.

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

Step 1.5.3

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

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Step 1.5.3.1

Cancel the common factor.

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

Step 1.5.3.2

Rewrite the expression.

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

Step 1.6

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Apply basic rules of exponents.

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Step 2.2.1.1

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

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

Step 2.2.1.2

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

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Step 2.2.1.2.1

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

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

Step 2.2.1.2.2

Multiply $2$ by $- 1$ .

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

Step 2.2.2

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

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

Step 2.2.3

Simplify.

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Step 2.2.3.1

Multiply $y$ by $y^{- 2}$ by adding the exponents.

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Step 2.2.3.1.1

Multiply $y$ by $y^{- 2}$ .

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Step 2.2.3.1.1.1

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

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

Step 2.2.3.1.1.2

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

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

Step 2.2.3.1.2

Subtract $2$ from $1$ .

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

Step 2.2.3.2

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

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

Step 2.2.4

Split the single integral into multiple integrals.

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

Step 2.2.5

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

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

Step 2.2.6

Since $- 1$ is constant with respect to $y$ , move $- 1$ out of the integral.

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

Step 2.2.7

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

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

Step 2.2.8

Simplify.

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Step 2.2.8.1

Simplify.

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

Step 2.2.8.2

Simplify.

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Step 2.2.8.2.1

Multiply $- 1$ by $- 1$ .

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

Step 2.2.8.2.2

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

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

Step 2.2.9

Reorder terms.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Apply basic rules of exponents.

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Step 2.3.1.1

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

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

Step 2.3.1.2

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

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Step 2.3.1.2.1

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

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

Step 2.3.1.2.2

Multiply $2$ by $- 1$ .

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

Step 2.3.2

Multiply $(1 + x) x^{- 2}$ .

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

Step 2.3.3

Simplify.

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Step 2.3.3.1

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

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

Step 2.3.3.2

Multiply $x$ by $x^{- 2}$ by adding the exponents.

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Step 2.3.3.2.1

Multiply $x$ by $x^{- 2}$ .

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Step 2.3.3.2.1.1

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

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

Step 2.3.3.2.1.2

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

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

Step 2.3.3.2.2

Subtract $2$ from $1$ .

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

Step 2.3.4

Split the single integral into multiple integrals.

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

Step 2.3.5

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

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

Step 2.3.6

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

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

Step 2.3.7

Simplify.

$\frac{1}{y} + \ln (| y |) + C_{3} = - \frac{1}{x} + \ln (| x |) + C_{6}$

Step 2.4

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

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