Calculus Examples

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

Step 1

Subtract $2 x^{2} d x$ from both sides of the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $- 2 (\frac{1}{3} x^{3} + C_{2})$ as $- 2 (\frac{1}{3}) x^{3} + C_{2}$ .

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

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

Combine $- 2$ and $\frac{1}{3}$ .

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

Step 2.3.3.2.2

Move the negative in front of the fraction.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $3$ .

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

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

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

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.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

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

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Step 3.2.2.1.1

Simplify each term.

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Step 3.2.2.1.1.1

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

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

Step 3.2.2.1.1.2

Move $2$ to the left of $x^{3}$ .

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

Step 3.2.2.1.2

Simplify terms.

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Step 3.2.2.1.2.1

Apply the distributive property.

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

Step 3.2.2.1.2.2

Cancel the common factor of $3$ .

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Step 3.2.2.1.2.2.1

Move the leading negative in $- \frac{2 x^{3}}{3}$ into the numerator.

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

Step 3.2.2.1.2.2.2

Cancel the common factor.

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

Step 3.2.2.1.2.2.3

Rewrite the expression.

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

Step 3.3

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

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

Step 4

Simplify the constant of integration.

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