Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Regroup factors.

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

Step 1.2

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

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

Step 1.3

Simplify.

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Step 1.3.1

Combine $\frac{x^{2}}{1 + x}$ and $y^{2}$ .

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

Step 1.3.2

Cancel the common factor of $1 + x$ .

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Step 1.3.2.1

Cancel the common factor.

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

Step 1.3.2.2

Rewrite the expression.

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

Step 1.3.3

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

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Step 1.3.3.1

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

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

Step 1.3.3.2

Cancel the common factor.

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

Step 1.3.3.3

Rewrite the expression.

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

Step 1.4

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

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

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

Step 2.2.1.2

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

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Step 2.2.1.2.1

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

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

Step 2.2.1.2.2

Multiply $2$ by $- 1$ .

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

Step 2.2.2

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

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

Step 2.2.3

Simplify.

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Step 2.2.3.1

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

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

Step 2.2.3.2

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

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Step 2.2.3.2.1

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

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Step 2.2.3.2.1.1

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

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

Step 2.2.3.2.1.2

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

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

Step 2.2.3.2.2

Subtract $2$ from $1$ .

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

Step 2.2.4

Split the single integral into multiple integrals.

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

Step 2.2.5

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

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

Step 2.2.6

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

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

Step 2.2.7

Simplify.

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

Step 2.3

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

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

Step 2.4

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

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