Calculus Examples

$\frac{d y}{d x} = \frac{x^{5}}{y}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $y$ .

$y \frac{d y}{d x} = y \frac{x^{5}}{y}$

Step 1.2

Cancel the common factor of $y$ .

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Step 1.2.1

Cancel the common factor.

$y \frac{d y}{d x} = y \frac{x^{5}}{y}$

Step 1.2.2

Rewrite the expression.

$y \frac{d y}{d x} = x^{5}$

Step 1.3

Rewrite the equation.

$y d y = x^{5} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y d y = \int x^{5} d x$

Step 2.2

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

$\frac{1}{2} y^{2} + C_{1} = \int x^{5} d x$

Step 2.3

By the Power Rule, the integral of $x^{5}$ with respect to $x$ is $\frac{1}{6} x^{6}$ .

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

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

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $2 (\frac{1}{2} y^{2})$ .

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Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $2 (\frac{1}{6} x^{6} + K)$ .

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Step 3.2.2.1.1

Combine $\frac{1}{6}$ and $x^{6}$ .

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

Step 3.2.2.1.2

Apply the distributive property.

$y^{2} = 2 \frac{x^{6}}{6} + 2 K$

Step 3.2.2.1.3

Cancel the common factor of $2$ .

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Step 3.2.2.1.3.1

Factor $2$ out of $6$ .

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

Step 3.2.2.1.3.2

Cancel the common factor.

$y^{2} = 2 \frac{x^{6}}{2 \cdot 3} + 2 K$

Step 3.2.2.1.3.3

Rewrite the expression.

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

Step 3.3

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

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

Step 3.4

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

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Step 3.4.1

To write $2 K$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 3.4.2

Combine $2 K$ and $\frac{3}{3}$ .

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

Step 3.4.3

Combine the numerators over the common denominator.

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

Step 3.4.4

Multiply $3$ by $2$ .

$y = \pm \sqrt{\frac{x^{6} + 6 K}{3}}$

Step 3.4.5

Rewrite $\sqrt{\frac{x^{6} + 6 K}{3}}$ as $\frac{\sqrt{x^{6} + 6 K}}{\sqrt{3}}$ .

$y = \pm \frac{\sqrt{x^{6} + 6 K}}{\sqrt{3}}$

Step 3.4.6

Multiply $\frac{\sqrt{x^{6} + 6 K}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 3.4.7

Combine and simplify the denominator.

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Step 3.4.7.1

Multiply $\frac{\sqrt{x^{6} + 6 K}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = \pm \frac{\sqrt{x^{6} + 6 K} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 3.4.7.2

Raise $\sqrt{3}$ to the power of $1$ .

$y = \pm \frac{\sqrt{x^{6} + 6 K} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 3.4.7.3

Raise $\sqrt{3}$ to the power of $1$ .

$y = \pm \frac{\sqrt{x^{6} + 6 K} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 3.4.7.4

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

$y = \pm \frac{\sqrt{x^{6} + 6 K} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 3.4.7.5

Add $1$ and $1$ .

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

Step 3.4.7.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.4.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$y = \pm \frac{\sqrt{x^{6} + 6 K} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 3.4.7.6.2

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

$y = \pm \frac{\sqrt{x^{6} + 6 K} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 3.4.7.6.3

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

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

Step 3.4.7.6.4

Cancel the common factor of $2$ .

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Step 3.4.7.6.4.1

Cancel the common factor.

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

Step 3.4.7.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{x^{6} + 6 K} \sqrt{3}}{3^{1}}$

Step 3.4.7.6.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{x^{6} + 6 K} \sqrt{3}}{3}$

Step 3.4.8

Combine using the product rule for radicals.

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

Step 3.4.9

Reorder factors in $\pm \frac{\sqrt{(x^{6} + 6 K) \cdot 3}}{3}$ .

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

Step 3.5

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

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Step 3.5.1

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

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

Step 3.5.2

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

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

Step 3.5.3

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

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

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

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

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

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

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

Step 4

Simplify the constant of integration.

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

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