Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $2 y - 4$ .

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

Step 1.2

Simplify.

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Step 1.2.1

Factor by grouping.

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Step 1.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 4 = - 12$ and whose sum is $b = 4$ .

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Step 1.2.1.1.1

Factor $4$ out of $4 x$ .

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

Step 1.2.1.1.2

Rewrite $4$ as $- 2$ plus $6$

$(2 y - 4) \frac{d y}{d x} = (2 y - 4) \frac{3 x^{2} + (- 2 + 6) x - 4}{2 y - 4}$

Step 1.2.1.1.3

Apply the distributive property.

$(2 y - 4) \frac{d y}{d x} = (2 y - 4) \frac{3 x^{2} - 2 x + 6 x - 4}{2 y - 4}$

Step 1.2.1.2

Factor out the greatest common factor from each group.

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Step 1.2.1.2.1

Group the first two terms and the last two terms.

$(2 y - 4) \frac{d y}{d x} = (2 y - 4) \frac{(3 x^{2} - 2 x) + 6 x - 4}{2 y - 4}$

Step 1.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 1.2.1.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 2$ .

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

Step 1.2.2

Factor $2$ out of $2 y - 4$ .

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Step 1.2.2.1

Factor $2$ out of $2 y$ .

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

Step 1.2.2.2

Factor $2$ out of $- 4$ .

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

Step 1.2.2.3

Factor $2$ out of $2 y + 2 \cdot - 2$ .

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

Step 1.2.3

Multiply $2 y - 4$ by $\frac{(3 x - 2) (x + 2)}{2 (y - 2)}$ .

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

Step 1.2.4

Cancel the common factor of $2 y - 4$ and $2$ .

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Step 1.2.4.1

Factor $2$ out of $(2 y - 4) ((3 x - 2) (x + 2))$ .

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

Step 1.2.4.2

Cancel the common factors.

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Step 1.2.4.2.1

Cancel the common factor.

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

Step 1.2.4.2.2

Rewrite the expression.

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

Step 1.2.5

Cancel the common factor of $y - 2$ .

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Step 1.2.5.1

Cancel the common factor.

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

Step 1.2.5.2

Divide $(3 x - 2) (x + 2)$ by $1$ .

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

Step 1.2.6

Expand $(3 x - 2) (x + 2)$ using the FOIL Method.

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Step 1.2.6.1

Apply the distributive property.

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

Step 1.2.6.2

Apply the distributive property.

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

Step 1.2.6.3

Apply the distributive property.

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

Step 1.2.7

Simplify and combine like terms.

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Step 1.2.7.1

Simplify each term.

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Step 1.2.7.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.2.7.1.1.1

Move $x$ .

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

Step 1.2.7.1.1.2

Multiply $x$ by $x$ .

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

Step 1.2.7.1.2

Multiply $2$ by $3$ .

$(2 y - 4) \frac{d y}{d x} = 3 x^{2} + 6 x - 2 x - 2 \cdot 2$

Step 1.2.7.1.3

Multiply $- 2$ by $2$ .

$(2 y - 4) \frac{d y}{d x} = 3 x^{2} + 6 x - 2 x - 4$

Step 1.2.7.2

Subtract $2 x$ from $6 x$ .

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

Step 1.3

Rewrite the equation.

$(2 y - 4) d y = (3 x^{2} + 4 x - 4) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Apply the constant rule.

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

Step 2.2.5

Simplify.

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Step 2.2.5.1

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

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

Step 2.2.5.2

Simplify.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

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

Step 2.3.3

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

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

Step 2.3.4

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

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

Step 2.3.5

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

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

Step 2.3.6

Apply the constant rule.

$y^{2} - 4 y + C_{3} = 3 (\frac{1}{3} x^{3} + C_{4}) + 4 (\frac{1}{2} x^{2} + C_{5}) - 4 x + C_{6}$

Step 2.3.7

Simplify.

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Step 2.3.7.1

Simplify.

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Step 2.3.7.1.1

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

$y^{2} - 4 y + C_{3} = 3 (\frac{x^{3}}{3} + C_{4}) + 4 (\frac{1}{2} x^{2} + C_{5}) - 4 x + C_{6}$

Step 2.3.7.1.2

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

$y^{2} - 4 y + C_{3} = 3 (\frac{x^{3}}{3} + C_{4}) + 4 (\frac{x^{2}}{2} + C_{5}) - 4 x + C_{6}$

Step 2.3.7.2

Simplify.

$y^{2} - 4 y + C_{3} = x^{3} + 2 x^{2} - 4 x + C_{7}$

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

Add $4 x$ to both sides of the equation.

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

Step 3.1.4

Subtract $K$ from both sides of the equation.

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

Step 3.2

Use the quadratic formula to find the solutions.

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

Step 3.3

Substitute the values $a = 1$ , $b = - 4$ , and $c = - x^{3} - 2 x^{2} + 4 x - K$ into the quadratic formula and solve for $y$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot (- x^{3} - 2 x^{2} + 4 x - K))}}{2 \cdot 1}$

Step 3.4

Simplify.

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Step 3.4.1

Simplify the numerator.

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Step 3.4.1.1

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot 1 \cdot (- x^{3} - 2 x^{2} + 4 x - K)$ .

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Step 3.4.1.1.1

Factor $- 4$ out of ${(- 4)}^{2}$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 \cdot 1 \cdot (- x^{3} - 2 x^{2} + 4 x - K)}}{2 \cdot 1}$

Step 3.4.1.1.2

Factor $- 4$ out of $- 4 \cdot 1 \cdot (- x^{3} - 2 x^{2} + 4 x - K)$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 (1 \cdot (- x^{3} - 2 x^{2} + 4 x - K))}}{2 \cdot 1}$

Step 3.4.1.1.3

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (1 \cdot (- x^{3} - 2 x^{2} + 4 x - K))$ .

$y = \frac{4 \pm \sqrt{- 4 (- 4 + 1 \cdot (- x^{3} - 2 x^{2} + 4 x - K))}}{2 \cdot 1}$

Step 3.4.1.2

Multiply $- x^{3} - 2 x^{2} + 4 x - K$ by $1$ .

$y = \frac{4 \pm \sqrt{- 4 (- 4 - x^{3} - 2 x^{2} + 4 x - K)}}{2 \cdot 1}$

Step 3.4.1.3

Rewrite $- 4 (- 4 - x^{3} - 2 x^{2} + 4 x - K)$ as $2^{2} \cdot (- 1 (- 4 - x^{3} - 2 x^{2} + 2^{2} x - K))$ .

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Step 3.4.1.3.1

Factor $4$ out of $- 4$ .

$y = \frac{4 \pm \sqrt{4 (- 1) (- 4 - x^{3} - 2 x^{2} + 4 x - K)}}{2 \cdot 1}$

Step 3.4.1.3.2

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

$y = \frac{4 \pm \sqrt{2^{2} \cdot (- 1 (- 4 - x^{3} - 2 x^{2} + 4 x - K))}}{2 \cdot 1}$

Step 3.4.1.3.3

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

$y = \frac{4 \pm \sqrt{2^{2} \cdot (- 1 (- 4 - x^{3} - 2 x^{2} + 2^{2} x - K))}}{2 \cdot 1}$

Step 3.4.1.3.4

Add parentheses.

$y = \frac{4 \pm \sqrt{2^{2} \cdot (- 1 (- 4 - x^{3} - 2 x^{2} + 2^{2} x - K))}}{2 \cdot 1}$

Step 3.4.1.4

Pull terms out from under the radical.

$y = \frac{4 \pm 2 \sqrt{- 1 (- 4 - x^{3} - 2 x^{2} + 2^{2} x - K)}}{2 \cdot 1}$

Step 3.4.1.5

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

$y = \frac{4 \pm 2 \sqrt{- 1 (- 4 - x^{3} - 2 x^{2} + 4 x - K)}}{2 \cdot 1}$

Step 3.4.2

Multiply $2$ by $1$ .

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

Step 3.4.3

Simplify $\frac{4 \pm 2 \sqrt{- 1 (- 4 - x^{3} - 2 x^{2} + 4 x - K)}}{2}$ .

$y = 2 \pm \sqrt{- 1 (- 4 - x^{3} - 2 x^{2} + 4 x - K)}$

Step 3.5

The final answer is the combination of both solutions.

$y = 2 + \sqrt{- 1 (- 4 - x^{3} - 2 x^{2} + 4 x - K)}$

$y = 2 - \sqrt{- 1 (- 4 - x^{3} - 2 x^{2} + 4 x - K)}$

$y = 2 + \sqrt{- 1 (- 4 - x^{3} - 2 x^{2} + 4 x - K)}$

$y = 2 - \sqrt{- 1 (- 4 - x^{3} - 2 x^{2} + 4 x - K)}$

Step 4

Simplify the constant of integration.

$y = 2 + \sqrt{- 1 (- 4 - x^{3} - 2 x^{2} + 4 x + K)}$

$y = 2 - \sqrt{- 1 (- 4 - x^{3} - 2 x^{2} + 4 x + K)}$