Calculus Examples

$8 y \frac{d y}{d x} = 9 x^{2}$

Step 1

Rewrite the equation.

$8 y d y = 9 x^{2} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int 8 y d y = \int 9 x^{2} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Since $8$ is constant with respect to $y$ , move $8$ out of the integral.

$8 \int y d y = \int 9 x^{2} d x$

Step 2.2.2

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

$8 (\frac{1}{2} y^{2} + C_{1}) = \int 9 x^{2} d x$

Step 2.2.3

Simplify the answer.

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Step 2.2.3.1

Rewrite $8 (\frac{1}{2} y^{2} + C_{1})$ as $8 (\frac{1}{2}) y^{2} + C_{1}$ .

$8 (\frac{1}{2}) y^{2} + C_{1} = \int 9 x^{2} d x$

Step 2.2.3.2

Simplify.

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Step 2.2.3.2.1

Combine $8$ and $\frac{1}{2}$ .

$\frac{8}{2} y^{2} + C_{1} = \int 9 x^{2} d x$

Step 2.2.3.2.2

Cancel the common factor of $8$ and $2$ .

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Step 2.2.3.2.2.1

Factor $2$ out of $8$ .

$\frac{2 \cdot 4}{2} y^{2} + C_{1} = \int 9 x^{2} d x$

Step 2.2.3.2.2.2

Cancel the common factors.

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Step 2.2.3.2.2.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 4}{2 (1)} y^{2} + C_{1} = \int 9 x^{2} d x$

Step 2.2.3.2.2.2.2

Cancel the common factor.

$\frac{2 \cdot 4}{2 \cdot 1} y^{2} + C_{1} = \int 9 x^{2} d x$

Step 2.2.3.2.2.2.3

Rewrite the expression.

$\frac{4}{1} y^{2} + C_{1} = \int 9 x^{2} d x$

Step 2.2.3.2.2.2.4

Divide $4$ by $1$ .

$4 y^{2} + C_{1} = \int 9 x^{2} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$4 y^{2} + C_{1} = 9 \int x^{2} d x$

Step 2.3.2

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

$4 y^{2} + C_{1} = 9 (\frac{1}{3} x^{3} + C_{2})$

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $9 (\frac{1}{3} x^{3} + C_{2})$ as $9 (\frac{1}{3}) x^{3} + C_{2}$ .

$4 y^{2} + C_{1} = 9 (\frac{1}{3}) x^{3} + C_{2}$

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

Combine $9$ and $\frac{1}{3}$ .

$4 y^{2} + C_{1} = \frac{9}{3} x^{3} + C_{2}$

Step 2.3.3.2.2

Cancel the common factor of $9$ and $3$ .

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Step 2.3.3.2.2.1

Factor $3$ out of $9$ .

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

Step 2.3.3.2.2.2

Cancel the common factors.

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Step 2.3.3.2.2.2.1

Factor $3$ out of $3$ .

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

Step 2.3.3.2.2.2.2

Cancel the common factor.

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

Step 2.3.3.2.2.2.3

Rewrite the expression.

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

Step 2.3.3.2.2.2.4

Divide $3$ by $1$ .

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Divide each term in $4 y^{2} = 3 x^{3} + K$ by $4$ and simplify.

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Step 3.1.1

Divide each term in $4 y^{2} = 3 x^{3} + K$ by $4$ .

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

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $4$ .

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Step 3.1.2.1.1

Cancel the common factor.

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

Step 3.1.2.1.2

Divide $y^{2}$ by $1$ .

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

Step 3.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{\frac{3 x^{3}}{4} + \frac{K}{4}}$

Step 3.3

Simplify $\pm \sqrt{\frac{3 x^{3}}{4} + \frac{K}{4}}$ .

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Step 3.3.1

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{3 x^{3} + K}{4}}$

Step 3.3.2

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

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Step 3.3.2.1

Factor the perfect power $1^{2}$ out of $3 x^{3} + K$ .

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

Step 3.3.2.2

Factor the perfect power $2^{2}$ out of $4$ .

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

Step 3.3.2.3

Rearrange the fraction $\frac{1^{2} (3 x^{3} + K)}{2^{2} \cdot 1}$ .

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

Step 3.3.3

Pull terms out from under the radical.

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

Step 3.3.4

Combine $\frac{1}{2}$ and $\sqrt{3 x^{3} + K}$ .

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

Step 3.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.4.1

First, use the positive value of the $\pm$ to find the first solution.

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

Step 3.4.2

Next, use the negative value of the $\pm$ to find the second solution.

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

Step 3.4.3

The complete solution is the result of both the positive and negative portions of the solution.

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

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

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

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

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

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