Calculus Examples

$8 y \frac{d y}{d x} = x$

Step 1

Rewrite the equation.

$8 y d y = x d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int 8 y d y = \int x d x$

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.

$8 \int y d y = \int x d x$

Step 2.2.2

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

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

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

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

Step 2.2.3.2

Simplify.

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Step 2.2.3.2.1

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

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

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 $8$ .

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

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 $2$ .

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

Step 2.2.3.2.2.2.2

Cancel the common factor.

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

Step 2.2.3.2.2.2.3

Rewrite the expression.

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

Step 2.2.3.2.2.2.4

Divide $4$ by $1$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Divide each term in $4 y^{2} = \frac{1}{2} x^{2} + K$ by $4$ and simplify.

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Step 3.1.1

Divide each term in $4 y^{2} = \frac{1}{2} x^{2} + K$ by $4$ .

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

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.

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

Step 3.1.2.1.2

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

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

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

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

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

Step 3.1.3.1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.1.3.1.3

Combine.

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

Step 3.1.3.1.4

Multiply $x^{2}$ by $1$ .

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

Step 3.1.3.1.5

Multiply $2$ by $4$ .

$y^{2} = \frac{x^{2}}{8} + \frac{K}{4}$

Step 3.2

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

$y = \pm \sqrt{\frac{x^{2}}{8} + \frac{K}{4}}$

Step 3.3

Simplify $\pm \sqrt{\frac{x^{2}}{8} + \frac{K}{4}}$ .

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Step 3.3.1

To write $\frac{K}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = \pm \sqrt{\frac{x^{2}}{8} + \frac{K}{4} \cdot \frac{2}{2}}$

Step 3.3.2

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.2.1

Multiply $\frac{K}{4}$ by $\frac{2}{2}$ .

$y = \pm \sqrt{\frac{x^{2}}{8} + \frac{K \cdot 2}{4 \cdot 2}}$

Step 3.3.2.2

Multiply $4$ by $2$ .

$y = \pm \sqrt{\frac{x^{2}}{8} + \frac{K \cdot 2}{8}}$

Step 3.3.3

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{x^{2} + K \cdot 2}{8}}$

Step 3.3.4

Move $2$ to the left of $K$ .

$y = \pm \sqrt{\frac{x^{2} + 2 K}{8}}$

Step 3.3.5

Rewrite $\frac{x^{2} + 2 K}{8}$ as ${(\frac{1}{2})}^{2} \frac{x^{2} + 2 K}{2}$ .

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Step 3.3.5.1

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

$y = \pm \sqrt{\frac{1^{2} (x^{2} + 2 K)}{8}}$

Step 3.3.5.2

Factor the perfect power $2^{2}$ out of $8$ .

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

Step 3.3.5.3

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

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

Step 3.3.6

Pull terms out from under the radical.

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

Step 3.3.7

Rewrite $\sqrt{\frac{x^{2} + 2 K}{2}}$ as $\frac{\sqrt{x^{2} + 2 K}}{\sqrt{2}}$ .

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

Step 3.3.8

Combine.

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

Step 3.3.9

Multiply $\sqrt{x^{2} + 2 K}$ by $1$ .

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

Step 3.3.10

Multiply $\frac{\sqrt{x^{2} + 2 K}}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 3.3.11

Combine and simplify the denominator.

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Step 3.3.11.1

Multiply $\frac{\sqrt{x^{2} + 2 K}}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 3.3.11.2

Move $\sqrt{2}$ .

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

Step 3.3.11.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 3.3.11.4

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 3.3.11.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.3.11.6

Add $1$ and $1$ .

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

Step 3.3.11.7

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 3.3.11.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Step 3.3.11.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.3.11.7.3

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

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

Step 3.3.11.7.4

Cancel the common factor of $2$ .

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Step 3.3.11.7.4.1

Cancel the common factor.

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

Step 3.3.11.7.4.2

Rewrite the expression.

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

Step 3.3.11.7.5

Evaluate the exponent.

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

Step 3.3.12

Combine using the product rule for radicals.

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

Step 3.3.13

Simplify the expression.

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Step 3.3.13.1

Multiply $2$ by $2$ .

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

Step 3.3.13.2

Reorder factors in $\pm \frac{\sqrt{(x^{2} + 2 K) \cdot 2}}{4}$ .

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

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.

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

Step 3.4.2

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

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

Step 3.4.3

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

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

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

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

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

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

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

Step 4

Simplify the constant of integration.

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

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