Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $y + 1$ .

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

Step 1.2

Simplify.

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Step 1.2.1

Factor $x$ out of $x^{2} + x$ .

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Step 1.2.1.1

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

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

Step 1.2.1.2

Raise $x$ to the power of $1$ .

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

Step 1.2.1.3

Factor $x$ out of $x^{1}$ .

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

Step 1.2.1.4

Factor $x$ out of $x \cdot x + x \cdot 1$ .

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

Step 1.2.2

Cancel the common factor of $y + 1$ .

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Step 1.2.2.1

Cancel the common factor.

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

Step 1.2.2.2

Rewrite the expression.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.2.4

Multiply $x$ by $x$ .

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

Step 1.2.5

Multiply $x$ by $1$ .

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

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

Step 2.2.2

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

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

Step 2.2.3

Apply the constant rule.

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

Step 2.2.4

Simplify.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

Simplify.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

Simplify each term.

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

Subtract $K$ from both sides of the equation.

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

Step 3.4

Multiply through by the least common denominator $6$ , then simplify.

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Step 3.4.1

Apply the distributive property.

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

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 $6$ .

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

Step 3.4.2.1.2

Cancel the common factor.

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

Step 3.4.2.1.3

Rewrite the expression.

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

Step 3.4.2.2

Cancel the common factor of $3$ .

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Step 3.4.2.2.1

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

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

Step 3.4.2.2.2

Factor $3$ out of $6$ .

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

Step 3.4.2.2.3

Cancel the common factor.

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

Step 3.4.2.2.4

Rewrite the expression.

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

Step 3.4.2.3

Multiply $- 1$ by $2$ .

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

Step 3.4.2.4

Cancel the common factor of $2$ .

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Step 3.4.2.4.1

Move the leading negative in $- \frac{x^{2}}{2}$ into the numerator.

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

Step 3.4.2.4.2

Factor $2$ out of $6$ .

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

Step 3.4.2.4.3

Cancel the common factor.

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

Step 3.4.2.4.4

Rewrite the expression.

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

Step 3.4.2.5

Multiply $- 1$ by $3$ .

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

Step 3.4.2.6

Multiply $- 1$ by $6$ .

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

Step 3.4.3

Move $6 y$ .

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

Step 3.4.4

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

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

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 $a = 3$ , $b = 6$ , and $c = - 2 x^{3} - 3 x^{2} - 6 K$ into the quadratic formula and solve for $y$ .

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

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 $6$ to the power of $2$ .

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

Step 3.7.1.2

Multiply $- 4$ by $3$ .

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

Step 3.7.1.3

Apply the distributive property.

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

Step 3.7.1.4

Simplify.

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Step 3.7.1.4.1

Multiply $- 2$ by $- 12$ .

$y = \frac{- 6 \pm \sqrt{36 + 24 x^{3} - 12 (- 3 x^{2}) - 12 (- 6 K)}}{2 \cdot 3}$

Step 3.7.1.4.2

Multiply $- 3$ by $- 12$ .

$y = \frac{- 6 \pm \sqrt{36 + 24 x^{3} + 36 x^{2} - 12 (- 6 K)}}{2 \cdot 3}$

Step 3.7.1.4.3

Multiply $- 6$ by $- 12$ .

$y = \frac{- 6 \pm \sqrt{36 + 24 x^{3} + 36 x^{2} + 72 K}}{2 \cdot 3}$

Step 3.7.1.5

Factor $12$ out of $36 + 24 x^{3} + 36 x^{2} + 72 K$ .

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Step 3.7.1.5.1

Factor $12$ out of $36$ .

$y = \frac{- 6 \pm \sqrt{12 (3) + 24 x^{3} + 36 x^{2} + 72 K}}{2 \cdot 3}$

Step 3.7.1.5.2

Factor $12$ out of $24 x^{3}$ .

$y = \frac{- 6 \pm \sqrt{12 (3) + 12 (2 x^{3}) + 36 x^{2} + 72 K}}{2 \cdot 3}$

Step 3.7.1.5.3

Factor $12$ out of $36 x^{2}$ .

$y = \frac{- 6 \pm \sqrt{12 (3) + 12 (2 x^{3}) + 12 (3 x^{2}) + 72 K}}{2 \cdot 3}$

Step 3.7.1.5.4

Factor $12$ out of $72 K$ .

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

Step 3.7.1.5.5

Factor $12$ out of $12 (3) + 12 (2 x^{3})$ .

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

Step 3.7.1.5.6

Factor $12$ out of $12 (3 + 2 x^{3}) + 12 (3 x^{2})$ .

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

Step 3.7.1.5.7

Factor $12$ out of $12 (3 + 2 x^{3} + 3 x^{2}) + 12 (6 K)$ .

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

Step 3.7.1.6

Rewrite $12 (3 + 2 x^{3} + 3 x^{2} + 6 K)$ as $2^{2} \cdot (3 (3 + 2 x^{3} + 3 x^{2} + 6 K))$ .

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Step 3.7.1.6.1

Factor $4$ out of $12$ .

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

Step 3.7.1.6.2

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

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

Step 3.7.1.6.3

Add parentheses.

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

Step 3.7.1.7

Pull terms out from under the radical.

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

Step 3.7.2

Multiply $2$ by $3$ .

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

Step 3.7.3

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

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

Step 3.8

The final answer is the combination of both solutions.

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

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

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

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

Step 4

Simplify the constant of integration.

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

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