Calculus Examples

$6 x^{2} d x - 2 (y d) y = 0$

Step 1

Subtract $6 x^{2} d x$ from both sides of the equation.

$- 2 y d y = - 6 x^{2} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int - 2 y d y = \int - 6 x^{2} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

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

$- 2 \int y d y = \int - 6 x^{2} d x$

Step 2.2.2

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

$- 2 (\frac{1}{2} y^{2} + C_{1}) = \int - 6 x^{2} d x$

Step 2.2.3

Simplify the answer.

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Step 2.2.3.1

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

$- 2 (\frac{1}{2}) y^{2} + C_{1} = \int - 6 x^{2} d x$

Step 2.2.3.2

Simplify.

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Step 2.2.3.2.1

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

$\frac{- 2}{2} y^{2} + C_{1} = \int - 6 x^{2} d x$

Step 2.2.3.2.2

Cancel the common factor of $- 2$ and $2$ .

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Step 2.2.3.2.2.1

Factor $2$ out of $- 2$ .

$\frac{2 \cdot - 1}{2} y^{2} + C_{1} = \int - 6 x^{2} d x$

Step 2.2.3.2.2.2

Cancel the common factors.

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Step 2.2.3.2.2.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot - 1}{2 (1)} y^{2} + C_{1} = \int - 6 x^{2} d x$

Step 2.2.3.2.2.2.2

Cancel the common factor.

$\frac{2 \cdot - 1}{2 \cdot 1} y^{2} + C_{1} = \int - 6 x^{2} d x$

Step 2.2.3.2.2.2.3

Rewrite the expression.

$\frac{- 1}{1} y^{2} + C_{1} = \int - 6 x^{2} d x$

Step 2.2.3.2.2.2.4

Divide $- 1$ by $1$ .

$- y^{2} + C_{1} = \int - 6 x^{2} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $- 6$ is constant with respect to $x$ , move $- 6$ out of the integral.

$- y^{2} + C_{1} = - 6 \int x^{2} d x$

Step 2.3.2

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

$- y^{2} + C_{1} = - 6 (\frac{1}{3} x^{3} + C_{2})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $- 6 (\frac{1}{3} x^{3} + C_{2})$ as $- 6 (\frac{1}{3}) x^{3} + C_{2}$ .

$- y^{2} + C_{1} = - 6 (\frac{1}{3}) x^{3} + C_{2}$

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

Combine $- 6$ and $\frac{1}{3}$ .

$- y^{2} + C_{1} = \frac{- 6}{3} x^{3} + C_{2}$

Step 2.3.3.2.2

Cancel the common factor of $- 6$ and $3$ .

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Step 2.3.3.2.2.1

Factor $3$ out of $- 6$ .

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

Step 2.3.3.2.2.2

Cancel the common factors.

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Step 2.3.3.2.2.2.1

Factor $3$ out of $3$ .

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

Step 2.3.3.2.2.2.2

Cancel the common factor.

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

Step 2.3.3.2.2.2.3

Rewrite the expression.

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

Step 2.3.3.2.2.2.4

Divide $- 2$ by $1$ .

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Divide each term in $- y^{2} = - 2 x^{3} + K$ by $- 1$ and simplify.

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Step 3.1.1

Divide each term in $- y^{2} = - 2 x^{3} + K$ by $- 1$ .

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

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Dividing two negative values results in a positive value.

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

Step 3.1.2.2

Divide $y^{2}$ by $1$ .

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

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

Move the negative one from the denominator of $\frac{- 2 x^{3}}{- 1}$ .

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

Step 3.1.3.1.2

Rewrite $- 1 \cdot (- 2 x^{3})$ as $- (- 2 x^{3})$ .

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

Step 3.1.3.1.3

Multiply $- 2$ by $- 1$ .

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

Step 3.1.3.1.4

Move the negative one from the denominator of $\frac{K}{- 1}$ .

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

Step 3.1.3.1.5

Rewrite $- 1 \cdot K$ as $- K$ .

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

Step 3.2

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

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

Step 3.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.3.1

First, use the positive value of the $\pm$ to find the first solution.

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

Step 3.3.2

Next, use the negative value of the $\pm$ to find the second solution.

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

Step 3.3.3

The complete solution is the result of both the positive and negative portions of the solution.

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

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

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

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

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

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

Step 4

Simplify the constant of integration.

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

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