Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $y + 3$ .

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

Step 1.2

Simplify.

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Step 1.2.1

Cancel the common factor of $y + 3$ .

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Step 1.2.1.1

Move the leading negative in $- \frac{x + 1}{2 (y + 3)}$ into the numerator.

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

Step 1.2.1.2

Factor $y + 3$ out of $2 (y + 3)$ .

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

Step 1.2.1.3

Cancel the common factor.

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

Step 1.2.1.4

Rewrite the expression.

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

Step 1.2.2

Move the negative in front of the fraction.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

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

Step 2.2.2

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

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

Step 2.2.3

Apply the constant rule.

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

Step 2.2.4

Simplify.

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

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.

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

Step 2.3.2

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

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

Step 2.3.3

Split the single integral into multiple integrals.

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

Step 2.3.4

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

$\frac{1}{2} y^{2} + 3 y + C_{3} = - \frac{1}{2} (\frac{1}{2} x^{2} + C_{4} + \int d x)$

Step 2.3.5

Apply the constant rule.

$\frac{1}{2} y^{2} + 3 y + C_{3} = - \frac{1}{2} (\frac{1}{2} x^{2} + C_{4} + x + C_{5})$

Step 2.3.6

Simplify.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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

Step 3.2.2

Apply the distributive property.

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

Step 3.2.3

Multiply $- \frac{1}{2} \cdot \frac{x^{2}}{2}$ .

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Step 3.2.3.1

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

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

Step 3.2.3.2

Multiply $2$ by $2$ .

$\frac{y^{2}}{2} + 3 y = - \frac{x^{2}}{4} - \frac{1}{2} x + K$

Step 3.2.4

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

$\frac{y^{2}}{2} + 3 y = - \frac{x^{2}}{4} - \frac{x}{2} + K$

Step 3.3

Move all the expressions to the left side of the equation.

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Step 3.3.1

Add $\frac{x^{2}}{4}$ to both sides of the equation.

$\frac{y^{2}}{2} + 3 y + \frac{x^{2}}{4} = - \frac{x}{2} + K$

Step 3.3.2

Add $\frac{x}{2}$ to both sides of the equation.

$\frac{y^{2}}{2} + 3 y + \frac{x^{2}}{4} + \frac{x}{2} = K$

Step 3.3.3

Subtract $K$ from both sides of the equation.

$\frac{y^{2}}{2} + 3 y + \frac{x^{2}}{4} + \frac{x}{2} - K = 0$

Step 3.4

Multiply through by the least common denominator $4$ , then simplify.

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Step 3.4.1

Apply the distributive property.

$4 (\frac{y^{2}}{2}) + 4 (3 y) + 4 (\frac{x^{2}}{4}) + 4 (\frac{x}{2}) + 4 (- K) = 0$

Step 3.4.2

Simplify.

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Step 3.4.2.1

Cancel the common factor of $2$ .

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Step 3.4.2.1.1

Factor $2$ out of $4$ .

$2 (2) (\frac{y^{2}}{2}) + 4 (3 y) + 4 (\frac{x^{2}}{4}) + 4 (\frac{x}{2}) + 4 (- K) = 0$

Step 3.4.2.1.2

Cancel the common factor.

$2 \cdot (2 (\frac{y^{2}}{2})) + 4 (3 y) + 4 (\frac{x^{2}}{4}) + 4 (\frac{x}{2}) + 4 (- K) = 0$

Step 3.4.2.1.3

Rewrite the expression.

$2 y^{2} + 4 (3 y) + 4 (\frac{x^{2}}{4}) + 4 (\frac{x}{2}) + 4 (- K) = 0$

Step 3.4.2.2

Multiply $3$ by $4$ .

$2 y^{2} + 12 y + 4 (\frac{x^{2}}{4}) + 4 (\frac{x}{2}) + 4 (- K) = 0$

Step 3.4.2.3

Cancel the common factor of $4$ .

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Step 3.4.2.3.1

Cancel the common factor.

$2 y^{2} + 12 y + 4 (\frac{x^{2}}{4}) + 4 (\frac{x}{2}) + 4 (- K) = 0$

Step 3.4.2.3.2

Rewrite the expression.

$2 y^{2} + 12 y + x^{2} + 4 (\frac{x}{2}) + 4 (- K) = 0$

Step 3.4.2.4

Cancel the common factor of $2$ .

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Step 3.4.2.4.1

Factor $2$ out of $4$ .

$2 y^{2} + 12 y + x^{2} + 2 (2) (\frac{x}{2}) + 4 (- K) = 0$

Step 3.4.2.4.2

Cancel the common factor.

$2 y^{2} + 12 y + x^{2} + 2 \cdot (2 (\frac{x}{2})) + 4 (- K) = 0$

Step 3.4.2.4.3

Rewrite the expression.

$2 y^{2} + 12 y + x^{2} + 2 x + 4 (- K) = 0$

Step 3.4.2.5

Multiply $- 1$ by $4$ .

$2 y^{2} + 12 y + x^{2} + 2 x - 4 K = 0$

Step 3.4.3

Move $12 y$ .

$2 y^{2} + x^{2} + 2 x - 4 K + 12 y = 0$

Step 3.4.4

Move $2 x$ .

$2 y^{2} + x^{2} - 4 K + 2 x + 12 y = 0$

Step 3.4.5

Reorder $2 y^{2}$ and $x^{2}$ .

$x^{2} + 2 y^{2} - 4 K + 2 x + 12 y = 0$

Step 3.5

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.6

Substitute the values $a = 2$ , $b = 12$ , and $c = x^{2} - 4 K + 2 x$ into the quadratic formula and solve for $y$ .

$\frac{- 12 \pm \sqrt{12^{2} - 4 \cdot (2 \cdot (x^{2} - 4 K + 2 x))}}{2 \cdot 2}$

Step 3.7

Simplify.

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Step 3.7.1

Simplify the numerator.

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Step 3.7.1.1

Raise $12$ to the power of $2$ .

$y = \frac{- 12 \pm \sqrt{144 - 4 \cdot 2 \cdot (x^{2} - 4 K + 2 x)}}{2 \cdot 2}$

Step 3.7.1.2

Multiply $- 4$ by $2$ .

$y = \frac{- 12 \pm \sqrt{144 - 8 \cdot (x^{2} - 4 K + 2 x)}}{2 \cdot 2}$

Step 3.7.1.3

Apply the distributive property.

$y = \frac{- 12 \pm \sqrt{144 - 8 x^{2} - 8 (- 4 K) - 8 (2 x)}}{2 \cdot 2}$

Step 3.7.1.4

Simplify.

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Step 3.7.1.4.1

Multiply $- 4$ by $- 8$ .

$y = \frac{- 12 \pm \sqrt{144 - 8 x^{2} + 32 K - 8 (2 x)}}{2 \cdot 2}$

Step 3.7.1.4.2

Multiply $2$ by $- 8$ .

$y = \frac{- 12 \pm \sqrt{144 - 8 x^{2} + 32 K - 16 x}}{2 \cdot 2}$

Step 3.7.1.5

Factor $8$ out of $144 - 8 x^{2} + 32 K - 16 x$ .

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Step 3.7.1.5.1

Factor $8$ out of $144$ .

$y = \frac{- 12 \pm \sqrt{8 (18) - 8 x^{2} + 32 K - 16 x}}{2 \cdot 2}$

Step 3.7.1.5.2

Factor $8$ out of $- 8 x^{2}$ .

$y = \frac{- 12 \pm \sqrt{8 (18) + 8 (- x^{2}) + 32 K - 16 x}}{2 \cdot 2}$

Step 3.7.1.5.3

Factor $8$ out of $32 K$ .

$y = \frac{- 12 \pm \sqrt{8 (18) + 8 (- x^{2}) + 8 (4 K) - 16 x}}{2 \cdot 2}$

Step 3.7.1.5.4

Factor $8$ out of $- 16 x$ .

$y = \frac{- 12 \pm \sqrt{8 (18) + 8 (- x^{2}) + 8 (4 K) + 8 (- 2 x)}}{2 \cdot 2}$

Step 3.7.1.5.5

Factor $8$ out of $8 (18) + 8 (- x^{2})$ .

$y = \frac{- 12 \pm \sqrt{8 (18 - x^{2}) + 8 (4 K) + 8 (- 2 x)}}{2 \cdot 2}$

Step 3.7.1.5.6

Factor $8$ out of $8 (18 - x^{2}) + 8 (4 K)$ .

$y = \frac{- 12 \pm \sqrt{8 (18 - x^{2} + 4 K) + 8 (- 2 x)}}{2 \cdot 2}$

Step 3.7.1.5.7

Factor $8$ out of $8 (18 - x^{2} + 4 K) + 8 (- 2 x)$ .

$y = \frac{- 12 \pm \sqrt{8 (18 - x^{2} + 4 K - 2 x)}}{2 \cdot 2}$

Step 3.7.1.6

Rewrite $8 (18 - x^{2} + 4 K - 2 x)$ as $2^{2} \cdot (2 (18 - x^{2} + 2^{2} K - 2 x))$ .

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Step 3.7.1.6.1

Factor $4$ out of $8$ .

$y = \frac{- 12 \pm \sqrt{4 (2) (18 - x^{2} + 4 K - 2 x)}}{2 \cdot 2}$

Step 3.7.1.6.2

Rewrite $4$ as $2^{2}$ .

$y = \frac{- 12 \pm \sqrt{2^{2} \cdot (2 (18 - x^{2} + 4 K - 2 x))}}{2 \cdot 2}$

Step 3.7.1.6.3

Rewrite $4$ as $2^{2}$ .

$y = \frac{- 12 \pm \sqrt{2^{2} \cdot (2 (18 - x^{2} + 2^{2} K - 2 x))}}{2 \cdot 2}$

Step 3.7.1.6.4

Add parentheses.

$y = \frac{- 12 \pm \sqrt{2^{2} \cdot (2 (18 - x^{2} + 2^{2} K - 2 x))}}{2 \cdot 2}$

Step 3.7.1.7

Pull terms out from under the radical.

$y = \frac{- 12 \pm 2 \sqrt{2 (18 - x^{2} + 2^{2} K - 2 x)}}{2 \cdot 2}$

Step 3.7.1.8

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

$y = \frac{- 12 \pm 2 \sqrt{2 (18 - x^{2} + 4 K - 2 x)}}{2 \cdot 2}$

Step 3.7.2

Multiply $2$ by $2$ .

$y = \frac{- 12 \pm 2 \sqrt{2 (18 - x^{2} + 4 K - 2 x)}}{4}$

Step 3.7.3

Simplify $\frac{- 12 \pm 2 \sqrt{2 (18 - x^{2} + 4 K - 2 x)}}{4}$ .

$y = \frac{- 6 \pm \sqrt{2 (18 - x^{2} + 4 K - 2 x)}}{2}$

Step 3.8

The final answer is the combination of both solutions.

$y = - \frac{6 - \sqrt{2 (18 - x^{2} + 4 K - 2 x)}}{2}$

$y = - \frac{\sqrt{2 (- x^{2} + 4 K - 2 x + 18)} + 6}{2}$

$y = - \frac{6 - \sqrt{2 (18 - x^{2} + 4 K - 2 x)}}{2}$

$y = - \frac{\sqrt{2 (- x^{2} + 4 K - 2 x + 18)} + 6}{2}$

Step 4

Simplify the constant of integration.

$y = - \frac{6 - \sqrt{2 (18 - x^{2} + K - 2 x)}}{2}$

$y = - \frac{\sqrt{2 (- x^{2} + K - 2 x + 18)} + 6}{2}$