Calculus Examples

$\frac{d y}{d x} = \frac{1}{12 y^{2} + 2}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $12 y^{2} + 2$ .

$(12 y^{2} + 2) \frac{d y}{d x} = (12 y^{2} + 2) \frac{1}{12 y^{2} + 2}$

Step 1.2

Simplify.

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Step 1.2.1

Factor $2$ out of $12 y^{2} + 2$ .

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Step 1.2.1.1

Factor $2$ out of $12 y^{2}$ .

$(12 y^{2} + 2) \frac{d y}{d x} = (12 y^{2} + 2) \frac{1}{2 (6 y^{2}) + 2}$

Step 1.2.1.2

Factor $2$ out of $2$ .

$(12 y^{2} + 2) \frac{d y}{d x} = (12 y^{2} + 2) \frac{1}{2 (6 y^{2}) + 2 (1)}$

Step 1.2.1.3

Factor $2$ out of $2 (6 y^{2}) + 2 (1)$ .

$(12 y^{2} + 2) \frac{d y}{d x} = (12 y^{2} + 2) \frac{1}{2 (6 y^{2} + 1)}$

Step 1.2.2

Multiply $12 y^{2} + 2$ by $\frac{1}{2 (6 y^{2} + 1)}$ .

$(12 y^{2} + 2) \frac{d y}{d x} = \frac{12 y^{2} + 2}{2 (6 y^{2} + 1)}$

Step 1.2.3

Cancel the common factor of $12 y^{2} + 2$ and $2$ .

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Step 1.2.3.1

Factor $2$ out of $12 y^{2}$ .

$(12 y^{2} + 2) \frac{d y}{d x} = \frac{2 (6 y^{2}) + 2}{2 (6 y^{2} + 1)}$

Step 1.2.3.2

Factor $2$ out of $2$ .

$(12 y^{2} + 2) \frac{d y}{d x} = \frac{2 (6 y^{2}) + 2 \cdot 1}{2 (6 y^{2} + 1)}$

Step 1.2.3.3

Factor $2$ out of $2 (6 y^{2}) + 2 (1)$ .

$(12 y^{2} + 2) \frac{d y}{d x} = \frac{2 (6 y^{2} + 1)}{2 (6 y^{2} + 1)}$

Step 1.2.3.4

Cancel the common factors.

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Step 1.2.3.4.1

Cancel the common factor.

$(12 y^{2} + 2) \frac{d y}{d x} = \frac{2 (6 y^{2} + 1)}{2 (6 y^{2} + 1)}$

Step 1.2.3.4.2

Rewrite the expression.

$(12 y^{2} + 2) \frac{d y}{d x} = \frac{6 y^{2} + 1}{6 y^{2} + 1}$

Step 1.2.4

Cancel the common factor of $6 y^{2} + 1$ .

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Step 1.2.4.1

Cancel the common factor.

$(12 y^{2} + 2) \frac{d y}{d x} = \frac{6 y^{2} + 1}{6 y^{2} + 1}$

Step 1.2.4.2

Rewrite the expression.

$(12 y^{2} + 2) \frac{d y}{d x} = 1$

Step 1.3

Rewrite the equation.

$(12 y^{2} + 2) d y = 1 d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int 12 y^{2} + 2 d y = \int d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Split the single integral into multiple integrals.

$\int 12 y^{2} d y + \int 2 d y = \int d x$

Step 2.2.2

Since $12$ is constant with respect to $y$ , move $12$ out of the integral.

$12 \int y^{2} d y + \int 2 d y = \int d x$

Step 2.2.3

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

$12 (\frac{1}{3} y^{3} + C_{1}) + \int 2 d y = \int d x$

Step 2.2.4

Apply the constant rule.

$12 (\frac{1}{3} y^{3} + C_{1}) + 2 y + C_{2} = \int d x$

Step 2.2.5

Simplify.

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Step 2.2.5.1

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

$12 (\frac{y^{3}}{3} + C_{1}) + 2 y + C_{2} = \int d x$

Step 2.2.5.2

Simplify.

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

Step 2.3

Apply the constant rule.

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

Step 2.4

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

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