Calculus Examples

$2 x d x + 6 d y = 0$

Step 1

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

$6 d y = - 2 x d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int 6 d y = \int - 2 x d x$

Step 2.2

Apply the constant rule.

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

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.

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

Step 2.3.2

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

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

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

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

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

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

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

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

Step 2.3.3.2.2

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

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Step 2.3.3.2.2.1

Factor $2$ out of $- 2$ .

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

Step 2.3.3.2.2.2

Cancel the common factors.

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Step 2.3.3.2.2.2.1

Factor $2$ out of $2$ .

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

Step 2.3.3.2.2.2.2

Cancel the common factor.

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

Step 2.3.3.2.2.2.3

Rewrite the expression.

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

Step 2.3.3.2.2.2.4

Divide $- 1$ by $1$ .

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

Step 2.4

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

$6 y = - x^{2} + K$

Step 3

Divide each term in $6 y = - x^{2} + K$ by $6$ and simplify.

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Step 3.1

Divide each term in $6 y = - x^{2} + K$ by $6$ .

$\frac{6 y}{6} = \frac{- x^{2}}{6} + \frac{K}{6}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $6$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{6 y}{6} = \frac{- x^{2}}{6} + \frac{K}{6}$

Step 3.2.1.2

Divide $y$ by $1$ .

$y = \frac{- x^{2}}{6} + \frac{K}{6}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Move the negative in front of the fraction.

$y = - \frac{x^{2}}{6} + \frac{K}{6}$

Step 4

Simplify the constant of integration.

$y = - \frac{x^{2}}{6} + K$