Calculus Examples

$x d x - y^{2} d y = 0$

Step 1

Subtract $x d x$ from both sides of the equation.

$- y^{2} d y = - x d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Rewrite $- (\frac{1}{3} y^{3} + C_{1})$ as $- \frac{1}{3} y^{3} + C_{1}$ .

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $- 3$ .

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

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

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

Step 3.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.2.1.1.2.1

Move the leading negative in $- \frac{y^{3}}{3}$ into the numerator.

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

Step 3.2.1.1.2.2

Factor $3$ out of $- 3$ .

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

Step 3.2.1.1.2.3

Cancel the common factor.

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

Step 3.2.1.1.2.4

Rewrite the expression.

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

Step 3.2.1.1.3

Multiply.

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Step 3.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

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

Step 3.2.1.1.3.2

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

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $- 3 (- \frac{1}{2} x^{2} + K)$ .

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Step 3.2.2.1.1

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

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

Step 3.2.2.1.2

Apply the distributive property.

$y^{3} = - 3 (- \frac{x^{2}}{2}) - 3 K$

Step 3.2.2.1.3

Multiply $- 3 (- \frac{x^{2}}{2})$ .

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Step 3.2.2.1.3.1

Multiply $- 1$ by $- 3$ .

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

Step 3.2.2.1.3.2

Combine $3$ and $\frac{x^{2}}{2}$ .

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

Step 3.3

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

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

Step 3.4

Simplify $\sqrt[3]{\frac{3 x^{2}}{2} - 3 K}$ .

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Step 3.4.1

Factor $3$ out of $\frac{3 x^{2}}{2} - 3 K$ .

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Step 3.4.1.1

Factor $3$ out of $\frac{3 x^{2}}{2}$ .

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

Step 3.4.1.2

Factor $3$ out of $- 3 K$ .

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

Step 3.4.1.3

Factor $3$ out of $3 \frac{x^{2}}{2} + 3 (- K)$ .

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

Step 3.4.2

To write $- K$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.4.3

Combine $- K$ and $\frac{2}{2}$ .

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

Step 3.4.4

Combine the numerators over the common denominator.

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

Step 3.4.5

Multiply $2$ by $- 1$ .

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

Step 3.4.6

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

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

Step 3.4.7

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

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

Step 3.4.8

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

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

Step 3.4.9

Combine and simplify the denominator.

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Step 3.4.9.1

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

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

Step 3.4.9.2

Raise $\sqrt[3]{2}$ to the power of $1$ .

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

Step 3.4.9.3

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

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

Step 3.4.9.4

Add $1$ and $2$ .

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

Step 3.4.9.5

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

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Step 3.4.9.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2}$ as $2^{\frac{1}{3}}$ .

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

Step 3.4.9.5.2

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

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

Step 3.4.9.5.3

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

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

Step 3.4.9.5.4

Cancel the common factor of $3$ .

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Step 3.4.9.5.4.1

Cancel the common factor.

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

Step 3.4.9.5.4.2

Rewrite the expression.

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

Step 3.4.9.5.5

Evaluate the exponent.

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

Step 3.4.10

Simplify the numerator.

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Step 3.4.10.1

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

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

Step 3.4.10.2

Raise $2$ to the power of $2$ .

$y = \frac{\sqrt[3]{3 (x^{2} - 2 K)} \sqrt[3]{4}}{2}$

Step 3.4.11

Simplify the numerator.

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Step 3.4.11.1

Combine using the product rule for radicals.

$y = \frac{\sqrt[3]{3 (x^{2} - 2 K) \cdot 4}}{2}$

Step 3.4.11.2

Multiply $4$ by $3$ .

$y = \frac{\sqrt[3]{12 (x^{2} - 2 K)}}{2}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt[3]{12 (x^{2} + K)}}{2}$