Calculus Examples

$(x^{2} + 1) d x + x^{2} y^{2} d y = 0$

Step 1

Subtract $(x^{2} + 1) d x$ from both sides of the equation.

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

Step 2

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

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

Step 3

Simplify.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Rewrite using the commutative property of multiplication.

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

Step 3.3

Apply the distributive property.

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

Step 3.4

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

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Step 3.4.1

Move the leading negative in $- \frac{1}{x^{2}}$ into the numerator.

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

Step 3.4.2

Cancel the common factor.

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

Step 3.4.3

Rewrite the expression.

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

Step 3.5

Multiply $- 1$ by $1$ .

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

Step 4

Integrate both sides.

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Step 4.1

Set up an integral on each side.

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

Step 4.2

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

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

Step 4.3

Integrate the right side.

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Step 4.3.1

Split the single integral into multiple integrals.

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

Step 4.3.2

Apply the constant rule.

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

Step 4.3.3

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

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

Step 4.3.4

Apply basic rules of exponents.

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Step 4.3.4.1

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

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

Step 4.3.4.2

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

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Step 4.3.4.2.1

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

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

Step 4.3.4.2.2

Multiply $2$ by $- 1$ .

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

Step 4.3.5

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

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

Step 4.3.6

Simplify.

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Step 4.3.6.1

Simplify.

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

Step 4.3.6.2

Simplify.

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Step 4.3.6.2.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.6.2.2

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

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

Step 4.4

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

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

Step 5

Solve for $y$ .

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Step 5.1

Multiply both sides of the equation by $3$ .

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

Step 5.2

Simplify both sides of the equation.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

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

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Step 5.2.1.1.1

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

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

Step 5.2.1.1.2

Cancel the common factor of $3$ .

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Step 5.2.1.1.2.1

Cancel the common factor.

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

Step 5.2.1.1.2.2

Rewrite the expression.

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

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Simplify $3 (- x + \frac{1}{x} + K)$ .

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Step 5.2.2.1.1

Apply the distributive property.

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

Step 5.2.2.1.2

Simplify.

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Step 5.2.2.1.2.1

Multiply $- 1$ by $3$ .

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

Step 5.2.2.1.2.2

Combine $3$ and $\frac{1}{x}$ .

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

Step 5.3

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

$y = \sqrt[3]{- 3 x + \frac{3}{x} + 3 K}$

Step 5.4

Simplify $\sqrt[3]{- 3 x + \frac{3}{x} + 3 K}$ .

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Step 5.4.1

Factor $3$ out of $- 3 x + \frac{3}{x} + 3 K$ .

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Step 5.4.1.1

Factor $3$ out of $- 3 x$ .

$y = \sqrt[3]{3 (- x) + \frac{3}{x} + 3 K}$

Step 5.4.1.2

Factor $3$ out of $\frac{3}{x}$ .

$y = \sqrt[3]{3 (- x) + 3 \frac{1}{x} + 3 K}$

Step 5.4.1.3

Factor $3$ out of $3 (- x) + 3 \frac{1}{x}$ .

$y = \sqrt[3]{3 (- x + \frac{1}{x}) + 3 K}$

Step 5.4.1.4

Factor $3$ out of $3 (- x + \frac{1}{x}) + 3 K$ .

$y = \sqrt[3]{3 (- x + \frac{1}{x} + K)}$

Step 5.4.2

To write $- x$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$y = \sqrt[3]{3 (- x \cdot \frac{x}{x} + \frac{1}{x} + K)}$

Step 5.4.3

Simplify terms.

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Step 5.4.3.1

Combine $- x$ and $\frac{x}{x}$ .

$y = \sqrt[3]{3 (\frac{- x \cdot x}{x} + \frac{1}{x} + K)}$

Step 5.4.3.2

Combine the numerators over the common denominator.

$y = \sqrt[3]{3 (\frac{- x \cdot x + 1}{x} + K)}$

Step 5.4.4

Simplify the numerator.

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Step 5.4.4.1

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

$y = \sqrt[3]{3 (\frac{- x \cdot x + 1^{2}}{x} + K)}$

Step 5.4.4.2

Rewrite $x \cdot x$ as $x^{2}$ .

$y = \sqrt[3]{3 (\frac{- x^{2} + 1^{2}}{x} + K)}$

Step 5.4.4.3

Reorder $- x^{2}$ and $1^{2}$ .

$y = \sqrt[3]{3 (\frac{1^{2} - x^{2}}{x} + K)}$

Step 5.4.4.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x$ .

$y = \sqrt[3]{3 (\frac{(1 + x) (1 - x)}{x} + K)}$

Step 5.4.5

To write $K$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$y = \sqrt[3]{3 (\frac{(1 + x) (1 - x)}{x} + \frac{K x}{x})}$

Step 5.4.6

Combine the numerators over the common denominator.

$y = \sqrt[3]{3 \frac{(1 + x) (1 - x) + K x}{x}}$

Step 5.4.7

Simplify the numerator.

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Step 5.4.7.1

Expand $(1 + x) (1 - x)$ using the FOIL Method.

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Step 5.4.7.1.1

Apply the distributive property.

$y = \sqrt[3]{3 \frac{1 (1 - x) + x (1 - x) + K x}{x}}$

Step 5.4.7.1.2

Apply the distributive property.

$y = \sqrt[3]{3 \frac{1 \cdot 1 + 1 (- x) + x (1 - x) + K x}{x}}$

Step 5.4.7.1.3

Apply the distributive property.

$y = \sqrt[3]{3 \frac{1 \cdot 1 + 1 (- x) + x \cdot 1 + x (- x) + K x}{x}}$

Step 5.4.7.2

Simplify and combine like terms.

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Step 5.4.7.2.1

Simplify each term.

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Step 5.4.7.2.1.1

Multiply $1$ by $1$ .

$y = \sqrt[3]{3 \frac{1 + 1 (- x) + x \cdot 1 + x (- x) + K x}{x}}$

Step 5.4.7.2.1.2

Multiply $- x$ by $1$ .

$y = \sqrt[3]{3 \frac{1 - x + x \cdot 1 + x (- x) + K x}{x}}$

Step 5.4.7.2.1.3

Multiply $x$ by $1$ .

$y = \sqrt[3]{3 \frac{1 - x + x + x (- x) + K x}{x}}$

Step 5.4.7.2.1.4

Rewrite using the commutative property of multiplication.

$y = \sqrt[3]{3 \frac{1 - x + x - x \cdot x + K x}{x}}$

Step 5.4.7.2.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 5.4.7.2.1.5.1

Move $x$ .

$y = \sqrt[3]{3 \frac{1 - x + x - (x \cdot x) + K x}{x}}$

Step 5.4.7.2.1.5.2

Multiply $x$ by $x$ .

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

Step 5.4.7.2.2

Add $- x$ and $x$ .

$y = \sqrt[3]{3 \frac{1 + 0 - x^{2} + K x}{x}}$

Step 5.4.7.2.3

Add $1$ and $0$ .

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

Step 5.4.8

Combine $3$ and $\frac{1 - x^{2} + K x}{x}$ .

$y = \sqrt[3]{\frac{3 (1 - x^{2} + K x)}{x}}$

Step 5.4.9

Rewrite $\sqrt[3]{\frac{3 (1 - x^{2} + K x)}{x}}$ as $\frac{\sqrt[3]{3 (1 - x^{2} + K x)}}{\sqrt[3]{x}}$ .

$y = \frac{\sqrt[3]{3 (1 - x^{2} + K x)}}{\sqrt[3]{x}}$

Step 5.4.10

Multiply $\frac{\sqrt[3]{3 (1 - x^{2} + K x)}}{\sqrt[3]{x}}$ by $\frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{2}}$ .

$y = \frac{\sqrt[3]{3 (1 - x^{2} + K x)}}{\sqrt[3]{x}} \cdot \frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{2}}$

Step 5.4.11

Combine and simplify the denominator.

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Step 5.4.11.1

Multiply $\frac{\sqrt[3]{3 (1 - x^{2} + K x)}}{\sqrt[3]{x}}$ by $\frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{2}}$ .

$y = \frac{\sqrt[3]{3 (1 - x^{2} + K x)} {\sqrt[3]{x}}^{2}}{\sqrt[3]{x} {\sqrt[3]{x}}^{2}}$

Step 5.4.11.2

Raise $\sqrt[3]{x}$ to the power of $1$ .

$y = \frac{\sqrt[3]{3 (1 - x^{2} + K x)} {\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{1} {\sqrt[3]{x}}^{2}}$

Step 5.4.11.3

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

$y = \frac{\sqrt[3]{3 (1 - x^{2} + K x)} {\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{1 + 2}}$

Step 5.4.11.4

Add $1$ and $2$ .

$y = \frac{\sqrt[3]{3 (1 - x^{2} + K x)} {\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{3}}$

Step 5.4.11.5

Rewrite ${\sqrt[3]{x}}^{3}$ as $x$ .

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Step 5.4.11.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$y = \frac{\sqrt[3]{3 (1 - x^{2} + K x)} {\sqrt[3]{x}}^{2}}{{(x^{\frac{1}{3}})}^{3}}$

Step 5.4.11.5.2

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

$y = \frac{\sqrt[3]{3 (1 - x^{2} + K x)} {\sqrt[3]{x}}^{2}}{x^{\frac{1}{3} \cdot 3}}$

Step 5.4.11.5.3

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

$y = \frac{\sqrt[3]{3 (1 - x^{2} + K x)} {\sqrt[3]{x}}^{2}}{x^{\frac{3}{3}}}$

Step 5.4.11.5.4

Cancel the common factor of $3$ .

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Step 5.4.11.5.4.1

Cancel the common factor.

$y = \frac{\sqrt[3]{3 (1 - x^{2} + K x)} {\sqrt[3]{x}}^{2}}{x^{\frac{3}{3}}}$

Step 5.4.11.5.4.2

Rewrite the expression.

$y = \frac{\sqrt[3]{3 (1 - x^{2} + K x)} {\sqrt[3]{x}}^{2}}{x^{1}}$

Step 5.4.11.5.5

Simplify.

$y = \frac{\sqrt[3]{3 (1 - x^{2} + K x)} {\sqrt[3]{x}}^{2}}{x}$

Step 5.4.12

Rewrite ${\sqrt[3]{x}}^{2}$ as $\sqrt[3]{x^{2}}$ .

$y = \frac{\sqrt[3]{3 (1 - x^{2} + K x)} \sqrt[3]{x^{2}}}{x}$

Step 5.4.13

Combine using the product rule for radicals.

$y = \frac{\sqrt[3]{3 (1 - x^{2} + K x) x^{2}}}{x}$

Step 5.4.14

Reorder factors in $\frac{\sqrt[3]{3 (1 - x^{2} + K x) x^{2}}}{x}$ .

$y = \frac{\sqrt[3]{3 x^{2} (1 - x^{2} + K x)}}{x}$