Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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Step 1.1.1

Divide each term in $x y^{2} \frac{d y}{d x} = x + 1$ by $x y^{2}$ .

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

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of $x$ .

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Step 1.1.2.1.1

Cancel the common factor.

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

Step 1.1.2.1.2

Rewrite the expression.

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

Step 1.1.2.2

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

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Step 1.1.2.2.1

Cancel the common factor.

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

Step 1.1.2.2.2

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

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

Step 1.1.3

Simplify the right side.

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Step 1.1.3.1

Cancel the common factor of $x$ .

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Step 1.1.3.1.1

Cancel the common factor.

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

Step 1.1.3.1.2

Rewrite the expression.

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

Step 1.2

Factor.

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Step 1.2.1

To write $\frac{1}{y^{2}}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 1.2.2

Write each expression with a common denominator of $x y^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.2.1

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

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

Step 1.2.2.2

Reorder the factors of $x y^{2}$ .

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

Step 1.2.3

Combine the numerators over the common denominator.

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

Step 1.3

Regroup factors.

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

Step 1.4

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

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

Step 1.5

Simplify.

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Step 1.5.1

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

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

Step 1.5.2

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

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Step 1.5.2.1

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

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

Step 1.5.2.2

Cancel the common factor.

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

Step 1.5.2.3

Rewrite the expression.

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

Step 1.6

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the fraction into multiple fractions.

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

Step 2.3.2

Split the single integral into multiple integrals.

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

Step 2.3.3

Cancel the common factor of $x$ .

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Step 2.3.3.1

Cancel the common factor.

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

Step 2.3.3.2

Rewrite the expression.

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

Step 2.3.4

Apply the constant rule.

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

Step 2.3.5

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

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

Step 2.3.6

Simplify.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $3$ .

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

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $3 (\frac{1}{3} y^{3})$ .

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Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{3} = 3 (x + \ln (| x |) + C)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Apply the distributive property.

$y^{3} = 3 x + 3 \ln (| x |) + 3 C$

Step 3.3

Simplify $3 \ln (| x |)$ by moving $3$ inside the logarithm.

$y^{3} = 3 x + \ln ({| x |}^{3}) + 3 C$

Step 3.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \sqrt[3]{3 x + \ln ({| x |}^{3}) + 3 C}$

Step 4

Simplify the constant of integration.

$y = \sqrt[3]{3 x + \ln ({| x |}^{3}) + C}$