Calculus Examples

$\frac{d y}{d x} = \frac{4 - 2 x}{3 y^{2} - 5}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $3 y^{2} - 5$ .

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

Step 1.2

Simplify.

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Step 1.2.1

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

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Step 1.2.1.1

Factor $2$ out of $4$ .

$(3 y^{2} - 5) \frac{d y}{d x} = (3 y^{2} - 5) \frac{2 (2) - 2 x}{3 y^{2} - 5}$

Step 1.2.1.2

Factor $2$ out of $- 2 x$ .

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

Step 1.2.1.3

Factor $2$ out of $2 (2) + 2 (- x)$ .

$(3 y^{2} - 5) \frac{d y}{d x} = (3 y^{2} - 5) \frac{2 (2 - x)}{3 y^{2} - 5}$

Step 1.2.2

Cancel the common factor of $3 y^{2} - 5$ .

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Step 1.2.2.1

Cancel the common factor.

$(3 y^{2} - 5) \frac{d y}{d x} = (3 y^{2} - 5) \frac{2 (2 - x)}{3 y^{2} - 5}$

Step 1.2.2.2

Rewrite the expression.

$(3 y^{2} - 5) \frac{d y}{d x} = 2 (2 - x)$

Step 1.2.3

Apply the distributive property.

$(3 y^{2} - 5) \frac{d y}{d x} = 2 \cdot 2 + 2 (- x)$

Step 1.2.4

Multiply $2$ by $2$ .

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

Step 1.2.5

Multiply $- 1$ by $2$ .

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

Step 1.3

Rewrite the equation.

$(3 y^{2} - 5) d y = (4 - 2 x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int 3 y^{2} - 5 d y = \int 4 - 2 x d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

$\int 3 y^{2} d y + \int - 5 d y = \int 4 - 2 x d x$

Step 2.2.2

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

$3 \int y^{2} d y + \int - 5 d y = \int 4 - 2 x d x$

Step 2.2.3

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

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

Step 2.2.4

Apply the constant rule.

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

Step 2.2.5

Simplify.

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Step 2.2.5.1

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

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

Step 2.2.5.2

Simplify.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

Apply the constant rule.

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

Simplify.

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Step 2.3.5.1

Simplify.

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

Step 2.3.5.2

Simplify.

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Step 2.3.5.2.1

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

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

Step 2.3.5.2.2

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

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Step 2.3.5.2.2.1

Factor $2$ out of $- 2$ .

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

Step 2.3.5.2.2.2

Cancel the common factors.

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Step 2.3.5.2.2.2.1

Factor $2$ out of $2$ .

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

Step 2.3.5.2.2.2.2

Cancel the common factor.

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

Step 2.3.5.2.2.2.3

Rewrite the expression.

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

Step 2.3.5.2.2.2.4

Divide $- 1$ by $1$ .

$y^{3} - 5 y + C_{3} = 4 x - x^{2} + C_{6}$

Step 2.3.6

Reorder terms.

$y^{3} - 5 y + C_{3} = - x^{2} + 4 x + C_{6}$

Step 2.4

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

$y^{3} - 5 y = - x^{2} + 4 x + K$