Calculus Examples

$\frac{d y}{d x} = \frac{x^{3} y^{3} - x^{2} y^{3}}{x + x y^{4}}$

Step 1

Separate the variables.

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Step 1.1

Factor $x^{2} y^{3}$ out of $x^{3} y^{3} - x^{2} y^{3}$ .

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Step 1.1.1

Factor $x^{2} y^{3}$ out of $x^{3} y^{3}$ .

$\frac{d y}{d x} = \frac{x^{2} y^{3} (x) - x^{2} y^{3}}{x + x y^{4}}$

Step 1.1.2

Factor $x^{2} y^{3}$ out of $- x^{2} y^{3}$ .

$\frac{d y}{d x} = \frac{x^{2} y^{3} (x) + x^{2} y^{3} (- 1)}{x + x y^{4}}$

Step 1.1.3

Factor $x^{2} y^{3}$ out of $x^{2} y^{3} (x) + x^{2} y^{3} (- 1)$ .

$\frac{d y}{d x} = \frac{x^{2} y^{3} (x - 1)}{x + x y^{4}}$

Step 1.2

Factor $x$ out of $x + x y^{4}$ .

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Step 1.2.1

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

$\frac{d y}{d x} = \frac{x^{2} y^{3} (x - 1)}{x^{1} + x y^{4}}$

Step 1.2.2

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

$\frac{d y}{d x} = \frac{x^{2} y^{3} (x - 1)}{x \cdot 1 + x y^{4}}$

Step 1.2.3

Factor $x$ out of $x y^{4}$ .

$\frac{d y}{d x} = \frac{x^{2} y^{3} (x - 1)}{x \cdot 1 + x (y^{4})}$

Step 1.2.4

Factor $x$ out of $x \cdot 1 + x (y^{4})$ .

$\frac{d y}{d x} = \frac{x^{2} y^{3} (x - 1)}{x \cdot (1 + y^{4})}$

Step 1.2.5

Multiply $x$ by $1 + y^{4}$ .

$\frac{d y}{d x} = \frac{x^{2} y^{3} (x - 1)}{x (1 + y^{4})}$

Step 1.3

Regroup factors.

$\frac{d y}{d x} = \frac{x^{2} (x - 1)}{x} \cdot \frac{y^{3}}{1 + y^{4}}$

Step 1.4

Multiply both sides by $\frac{1 + y^{4}}{y^{3}}$ .

$\frac{1 + y^{4}}{y^{3}} \frac{d y}{d x} = \frac{1 + y^{4}}{y^{3}} (\frac{x^{2} (x - 1)}{x} \cdot \frac{y^{3}}{1 + y^{4}})$

Step 1.5

Simplify.

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Step 1.5.1

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

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Step 1.5.1.1

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

$\frac{1 + y^{4}}{y^{3}} \frac{d y}{d x} = \frac{1 + y^{4}}{y^{3}} (\frac{x (x (x - 1))}{x} \cdot \frac{y^{3}}{1 + y^{4}})$

Step 1.5.1.2

Cancel the common factors.

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Step 1.5.1.2.1

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

$\frac{1 + y^{4}}{y^{3}} \frac{d y}{d x} = \frac{1 + y^{4}}{y^{3}} (\frac{x (x (x - 1))}{x^{1}} \cdot \frac{y^{3}}{1 + y^{4}})$

Step 1.5.1.2.2

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

$\frac{1 + y^{4}}{y^{3}} \frac{d y}{d x} = \frac{1 + y^{4}}{y^{3}} (\frac{x (x (x - 1))}{x \cdot 1} \cdot \frac{y^{3}}{1 + y^{4}})$

Step 1.5.1.2.3

Cancel the common factor.

$\frac{1 + y^{4}}{y^{3}} \frac{d y}{d x} = \frac{1 + y^{4}}{y^{3}} (\frac{x (x (x - 1))}{x \cdot 1} \cdot \frac{y^{3}}{1 + y^{4}})$

Step 1.5.1.2.4

Rewrite the expression.

$\frac{1 + y^{4}}{y^{3}} \frac{d y}{d x} = \frac{1 + y^{4}}{y^{3}} (\frac{x (x - 1)}{1} \cdot \frac{y^{3}}{1 + y^{4}})$

Step 1.5.1.2.5

Divide $x (x - 1)$ by $1$ .

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

Step 1.5.2

Apply the distributive property.

$\frac{1 + y^{4}}{y^{3}} \frac{d y}{d x} = \frac{1 + y^{4}}{y^{3}} ((x \cdot x + x \cdot - 1) \frac{y^{3}}{1 + y^{4}})$

Step 1.5.3

Multiply $x$ by $x$ .

$\frac{1 + y^{4}}{y^{3}} \frac{d y}{d x} = \frac{1 + y^{4}}{y^{3}} ((x^{2} + x \cdot - 1) \frac{y^{3}}{1 + y^{4}})$

Step 1.5.4

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

$\frac{1 + y^{4}}{y^{3}} \frac{d y}{d x} = \frac{1 + y^{4}}{y^{3}} ((x^{2} - 1 \cdot x) \frac{y^{3}}{1 + y^{4}})$

Step 1.5.5

Rewrite $- 1 x$ as $- x$ .

$\frac{1 + y^{4}}{y^{3}} \frac{d y}{d x} = \frac{1 + y^{4}}{y^{3}} ((x^{2} - x) \frac{y^{3}}{1 + y^{4}})$

Step 1.5.6

Multiply $x^{2} - x$ by $\frac{y^{3}}{1 + y^{4}}$ .

$\frac{1 + y^{4}}{y^{3}} \frac{d y}{d x} = \frac{1 + y^{4}}{y^{3}} \cdot \frac{(x^{2} - x) y^{3}}{1 + y^{4}}$

Step 1.5.7

Cancel the common factor of $1 + y^{4}$ .

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Step 1.5.7.1

Cancel the common factor.

$\frac{1 + y^{4}}{y^{3}} \frac{d y}{d x} = \frac{1 + y^{4}}{y^{3}} \cdot \frac{(x^{2} - x) y^{3}}{1 + y^{4}}$

Step 1.5.7.2

Rewrite the expression.

$\frac{1 + y^{4}}{y^{3}} \frac{d y}{d x} = \frac{1}{y^{3}} ((x^{2} - x) y^{3})$

Step 1.5.8

Cancel the common factor of $y^{3}$ .

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Step 1.5.8.1

Factor $y^{3}$ out of $(x^{2} - x) y^{3}$ .

$\frac{1 + y^{4}}{y^{3}} \frac{d y}{d x} = \frac{1}{y^{3}} (y^{3} (x^{2} - x))$

Step 1.5.8.2

Cancel the common factor.

$\frac{1 + y^{4}}{y^{3}} \frac{d y}{d x} = \frac{1}{y^{3}} (y^{3} (x^{2} - x))$

Step 1.5.8.3

Rewrite the expression.

$\frac{1 + y^{4}}{y^{3}} \frac{d y}{d x} = x^{2} - x$

Step 1.6

Rewrite the equation.

$\frac{1 + y^{4}}{y^{3}} d y = (x^{2} - x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1 + y^{4}}{y^{3}} d y = \int x^{2} - x d x$

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 $y^{3}$ out of the denominator by raising it to the $- 1$ power.

$\int (1 + y^{4}) {(y^{3})}^{- 1} d y = \int x^{2} - x d x$

Step 2.2.1.2

Multiply the exponents in ${(y^{3})}^{- 1}$ .

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Step 2.2.1.2.1

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

$\int (1 + y^{4}) y^{3 \cdot - 1} d y = \int x^{2} - x d x$

Step 2.2.1.2.2

Multiply $3$ by $- 1$ .

$\int (1 + y^{4}) y^{- 3} d y = \int x^{2} - x d x$

Step 2.2.2

Multiply $(1 + y^{4}) y^{- 3}$ .

$\int 1 y^{- 3} + y^{4} y^{- 3} d y = \int x^{2} - x d x$

Step 2.2.3

Simplify.

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Step 2.2.3.1

Multiply $y^{- 3}$ by $1$ .

$\int y^{- 3} + y^{4} y^{- 3} d y = \int x^{2} - x d x$

Step 2.2.3.2

Multiply $y^{4}$ by $y^{- 3}$ by adding the exponents.

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Step 2.2.3.2.1

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

$\int y^{- 3} + y^{4 - 3} d y = \int x^{2} - x d x$

Step 2.2.3.2.2

Subtract $3$ from $4$ .

$\int y^{- 3} + y^{1} d y = \int x^{2} - x d x$

Step 2.2.3.3

Simplify $y^{1}$ .

$\int y^{- 3} + y d y = \int x^{2} - x d x$

Step 2.2.4

Split the single integral into multiple integrals.

$\int y^{- 3} d y + \int y d y = \int x^{2} - x d x$

Step 2.2.5

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

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

Step 2.2.6

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

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

Step 2.2.7

Simplify.

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Step 2.2.7.1

Simplify.

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

Step 2.2.7.2

Simplify.

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Step 2.2.7.2.1

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

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

Step 2.2.7.2.2

Move $2$ to the left of $y^{2}$ .

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

Step 2.2.8

Reorder terms.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

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

$\frac{1}{2} y^{2} - \frac{1}{2 y^{2}} + C_{3} = \frac{1}{3} x^{3} + C_{4} + \int - x d x$

Step 2.3.3

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

$\frac{1}{2} y^{2} - \frac{1}{2 y^{2}} + C_{3} = \frac{1}{3} x^{3} + C_{4} - \int x d x$

Step 2.3.4

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

$\frac{1}{2} y^{2} - \frac{1}{2 y^{2}} + C_{3} = \frac{1}{3} x^{3} + C_{4} - (\frac{1}{2} x^{2} + C_{5})$

Step 2.3.5

Simplify.

$\frac{1}{2} y^{2} - \frac{1}{2 y^{2}} + C_{3} = \frac{1}{3} x^{3} - \frac{1}{2} x^{2} + C_{6}$

Step 2.4

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

$\frac{1}{2} y^{2} - \frac{1}{2 y^{2}} = \frac{1}{3} x^{3} - \frac{1}{2} x^{2} + K$