Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $2 - 2 y$ .

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

Step 1.2

Simplify.

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Step 1.2.1

Factor $2$ out of $2 - 2 y$ .

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Step 1.2.1.1

Factor $2$ out of $2$ .

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

Step 1.2.1.2

Factor $2$ out of $- 2 y$ .

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

Step 1.2.1.3

Factor $2$ out of $2 (1) + 2 (- y)$ .

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

Step 1.2.2

Multiply $2 - 2 y$ by $\frac{x^{2} + 1}{2 (1 - y)}$ .

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

Step 1.2.3

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

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Step 1.2.3.1

Factor $2$ out of $(2 - 2 y) (x^{2} + 1)$ .

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

Step 1.2.3.2

Cancel the common factors.

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Step 1.2.3.2.1

Cancel the common factor.

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

Step 1.2.3.2.2

Rewrite the expression.

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

Step 1.2.4

Cancel the common factor of $1 - y$ .

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Step 1.2.4.1

Cancel the common factor.

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

Step 1.2.4.2

Divide $x^{2} + 1$ by $1$ .

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

Step 1.3

Rewrite the equation.

$(2 - 2 y) d y = (x^{2} + 1) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

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

Step 2.2.2

Apply the constant rule.

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Simplify.

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Step 2.2.5.1

Simplify.

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

Step 2.2.5.2

Simplify.

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Step 2.2.5.2.1

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

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

Step 2.2.5.2.2

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

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Step 2.2.5.2.2.1

Factor $2$ out of $- 2$ .

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

Step 2.2.5.2.2.2

Cancel the common factors.

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Step 2.2.5.2.2.2.1

Factor $2$ out of $2$ .

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

Step 2.2.5.2.2.2.2

Cancel the common factor.

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

Step 2.2.5.2.2.2.3

Rewrite the expression.

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

Step 2.2.5.2.2.2.4

Divide $- 1$ by $1$ .

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

Step 2.2.6

Reorder terms.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

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

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

Step 2.3.3

Apply the constant rule.

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

Step 2.3.4

Simplify.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

Subtract $\frac{x^{3}}{3}$ from both sides of the equation.

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

Step 3.2.2

Subtract $x$ from both sides of the equation.

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

Step 3.2.3

Subtract $K$ from both sides of the equation.

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

Step 3.3

Multiply through by the least common denominator $3$ , then simplify.

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Step 3.3.1

Apply the distributive property.

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

Step 3.3.2

Simplify.

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Step 3.3.2.1

Multiply $- 1$ by $3$ .

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

Step 3.3.2.2

Multiply $2$ by $3$ .

$- 3 y^{2} + 6 y + 3 (- \frac{x^{3}}{3}) + 3 (- x) + 3 (- K) = 0$

Step 3.3.2.3

Cancel the common factor of $3$ .

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Step 3.3.2.3.1

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

$- 3 y^{2} + 6 y + 3 (\frac{- x^{3}}{3}) + 3 (- x) + 3 (- K) = 0$

Step 3.3.2.3.2

Cancel the common factor.

$- 3 y^{2} + 6 y + 3 (\frac{- x^{3}}{3}) + 3 (- x) + 3 (- K) = 0$

Step 3.3.2.3.3

Rewrite the expression.

$- 3 y^{2} + 6 y - x^{3} + 3 (- x) + 3 (- K) = 0$

Step 3.3.2.4

Multiply $- 1$ by $3$ .

$- 3 y^{2} + 6 y - x^{3} - 3 x + 3 (- K) = 0$

Step 3.3.2.5

Multiply $- 1$ by $3$ .

$- 3 y^{2} + 6 y - x^{3} - 3 x - 3 K = 0$

Step 3.3.3

Move $- 3 x$ .

$- 3 y^{2} + 6 y - x^{3} - 3 K - 3 x = 0$

Step 3.3.4

Move $6 y$ .

$- 3 y^{2} - x^{3} - 3 K + 6 y - 3 x = 0$

Step 3.3.5

Reorder $- 3 y^{2}$ and $- x^{3}$ .

$- x^{3} - 3 y^{2} - 3 K + 6 y - 3 x = 0$

Step 3.4

Use the quadratic formula to find the solutions.

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

Step 3.5

Substitute the values $a = - 3$ , $b = 6$ , and $c = - x^{3} - 3 K - 3 x$ into the quadratic formula and solve for $y$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (- 3 \cdot (- x^{3} - 3 K - 3 x))}}{2 \cdot - 3}$

Step 3.6

Simplify.

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

Raise $6$ to the power of $2$ .

$y = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 3 \cdot (- x^{3} - 3 K - 3 x)}}{2 \cdot - 3}$

Step 3.6.1.2

Multiply $- 4$ by $- 3$ .

$y = \frac{- 6 \pm \sqrt{36 + 12 \cdot (- x^{3} - 3 K - 3 x)}}{2 \cdot - 3}$

Step 3.6.1.3

Apply the distributive property.

$y = \frac{- 6 \pm \sqrt{36 + 12 (- x^{3}) + 12 (- 3 K) + 12 (- 3 x)}}{2 \cdot - 3}$

Step 3.6.1.4

Simplify.

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Step 3.6.1.4.1

Multiply $- 1$ by $12$ .

$y = \frac{- 6 \pm \sqrt{36 - 12 x^{3} + 12 (- 3 K) + 12 (- 3 x)}}{2 \cdot - 3}$

Step 3.6.1.4.2

Multiply $- 3$ by $12$ .

$y = \frac{- 6 \pm \sqrt{36 - 12 x^{3} - 36 K + 12 (- 3 x)}}{2 \cdot - 3}$

Step 3.6.1.4.3

Multiply $- 3$ by $12$ .

$y = \frac{- 6 \pm \sqrt{36 - 12 x^{3} - 36 K - 36 x}}{2 \cdot - 3}$

Step 3.6.1.5

Factor $12$ out of $36 - 12 x^{3} - 36 K - 36 x$ .

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Step 3.6.1.5.1

Factor $12$ out of $36$ .

$y = \frac{- 6 \pm \sqrt{12 (3) - 12 x^{3} - 36 K - 36 x}}{2 \cdot - 3}$

Step 3.6.1.5.2

Factor $12$ out of $- 12 x^{3}$ .

$y = \frac{- 6 \pm \sqrt{12 (3) + 12 (- x^{3}) - 36 K - 36 x}}{2 \cdot - 3}$

Step 3.6.1.5.3

Factor $12$ out of $- 36 K$ .

$y = \frac{- 6 \pm \sqrt{12 (3) + 12 (- x^{3}) + 12 (- 3 K) - 36 x}}{2 \cdot - 3}$

Step 3.6.1.5.4

Factor $12$ out of $- 36 x$ .

$y = \frac{- 6 \pm \sqrt{12 (3) + 12 (- x^{3}) + 12 (- 3 K) + 12 (- 3 x)}}{2 \cdot - 3}$

Step 3.6.1.5.5

Factor $12$ out of $12 (3) + 12 (- x^{3})$ .

$y = \frac{- 6 \pm \sqrt{12 (3 - x^{3}) + 12 (- 3 K) + 12 (- 3 x)}}{2 \cdot - 3}$

Step 3.6.1.5.6

Factor $12$ out of $12 (3 - x^{3}) + 12 (- 3 K)$ .

$y = \frac{- 6 \pm \sqrt{12 (3 - x^{3} - 3 K) + 12 (- 3 x)}}{2 \cdot - 3}$

Step 3.6.1.5.7

Factor $12$ out of $12 (3 - x^{3} - 3 K) + 12 (- 3 x)$ .

$y = \frac{- 6 \pm \sqrt{12 (3 - x^{3} - 3 K - 3 x)}}{2 \cdot - 3}$

Step 3.6.1.6

Rewrite $12 (3 - x^{3} - 3 K - 3 x)$ as $2^{2} \cdot (3 (3 - x^{3} - 3 K - 3 x))$ .

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Step 3.6.1.6.1

Factor $4$ out of $12$ .

$y = \frac{- 6 \pm \sqrt{4 (3) (3 - x^{3} - 3 K - 3 x)}}{2 \cdot - 3}$

Step 3.6.1.6.2

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

$y = \frac{- 6 \pm \sqrt{2^{2} \cdot (3 (3 - x^{3} - 3 K - 3 x))}}{2 \cdot - 3}$

Step 3.6.1.6.3

Add parentheses.

$y = \frac{- 6 \pm \sqrt{2^{2} \cdot (3 (3 - x^{3} - 3 K - 3 x))}}{2 \cdot - 3}$

Step 3.6.1.7

Pull terms out from under the radical.

$y = \frac{- 6 \pm 2 \sqrt{3 (3 - x^{3} - 3 K - 3 x)}}{2 \cdot - 3}$

Step 3.6.2

Multiply $2$ by $- 3$ .

$y = \frac{- 6 \pm 2 \sqrt{3 (3 - x^{3} - 3 K - 3 x)}}{- 6}$

Step 3.6.3

Simplify $\frac{- 6 \pm 2 \sqrt{3 (3 - x^{3} - 3 K - 3 x)}}{- 6}$ .

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

Step 3.7

The final answer is the combination of both solutions.

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

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

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

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

Step 4

Simplify the constant of integration.

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

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