Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

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

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

Step 1.2

Regroup factors.

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

Step 1.3

Regroup factors.

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

Step 1.4

Multiply both sides by $y (1 + 2 y^{2})$ .

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

Step 1.5

Simplify.

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Step 1.5.1

Apply the distributive property.

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

Step 1.5.2

Multiply $y$ by $1$ .

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

Step 1.5.3

Rewrite using the commutative property of multiplication.

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

Step 1.5.4

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

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Step 1.5.4.1

Move $y^{2}$ .

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

Step 1.5.4.2

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

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Step 1.5.4.2.1

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

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

Step 1.5.4.2.2

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

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

Step 1.5.4.3

Add $2$ and $1$ .

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

Step 1.5.5

Combine.

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

Step 1.5.6

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

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

Step 1.5.7

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

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Step 1.5.7.1

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

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

Step 1.5.7.2

Cancel the common factors.

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Step 1.5.7.2.1

Factor $x$ out of $4 x y (1 + 2 y^{2})$ .

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

Step 1.5.7.2.2

Cancel the common factor.

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

Step 1.5.7.2.3

Rewrite the expression.

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

Step 1.5.8

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

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

Step 1.5.9

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

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Step 1.5.9.1

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

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

Step 1.5.9.2

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

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

Step 1.5.9.3

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

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

Step 1.5.9.4

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

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

Step 1.5.10

Cancel the common factor of $y$ .

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Step 1.5.10.1

Cancel the common factor.

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

Step 1.5.10.2

Rewrite the expression.

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

Step 1.5.11

Cancel the common factor of $1 + 2 y^{2}$ .

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Step 1.5.11.1

Cancel the common factor.

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

Step 1.5.11.2

Rewrite the expression.

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

Step 1.6

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Simplify.

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Step 2.2.1.1

Apply the distributive property.

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

Step 2.2.1.2

Reorder $y$ and $1$ .

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

Step 2.2.1.3

Reorder $y$ and $2$ .

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

Step 2.2.1.4

Multiply $y$ by $1$ .

$\int y + 2 y \cdot y^{2} d y = \int \frac{x}{4} d x$

Step 2.2.1.5

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

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

Step 2.2.1.6

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

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

Step 2.2.1.7

Add $1$ and $2$ .

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

Step 2.2.1.8

Reorder $y$ and $2 y^{3}$ .

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

Step 2.2.2

Split the single integral into multiple integrals.

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

Step 2.2.3

Since $2$ is constant with respect to $y$ , move $2$ out of the integral.

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

Step 2.2.4

By the Power Rule, the integral of $y^{3}$ with respect to $y$ is $\frac{1}{4} y^{4}$ .

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

Step 2.2.5

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

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

Step 2.2.6

Simplify.

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Step 2.2.6.1

Combine $\frac{1}{4}$ and $y^{4}$ .

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

Step 2.2.6.2

Simplify.

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

Step 2.2.6.3

Reorder terms.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $\frac{1}{4}$ is constant with respect to $x$ , move $\frac{1}{4}$ out of the integral.

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

Step 2.3.2

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

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

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $\frac{1}{4} (\frac{1}{2} x^{2} + C_{4})$ as $\frac{1}{4} \cdot \frac{1}{2} x^{2} + C_{4}$ .

$\frac{1}{2} y^{4} + \frac{1}{2} y^{2} + C_{3} = \frac{1}{4} \cdot \frac{1}{2} x^{2} + C_{4}$

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

Multiply $\frac{1}{4}$ by $\frac{1}{2}$ .

$\frac{1}{2} y^{4} + \frac{1}{2} y^{2} + C_{3} = \frac{1}{4 \cdot 2} x^{2} + C_{4}$

Step 2.3.3.2.2

Multiply $4$ by $2$ .

$\frac{1}{2} y^{4} + \frac{1}{2} y^{2} + C_{3} = \frac{1}{8} x^{2} + C_{4}$

Step 2.4

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

$\frac{1}{2} y^{4} + \frac{1}{2} y^{2} = \frac{1}{8} x^{2} + K$