Calculus Examples

$(4 + x^{6}) \frac{d y}{d x} = \frac{x^{5}}{y}$

Step 1

Separate the variables.

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Step 1.1

Divide each term in $(4 + x^{6}) \frac{d y}{d x} = \frac{x^{5}}{y}$ by $4 + x^{6}$ and simplify.

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Step 1.1.1

Divide each term in $(4 + x^{6}) \frac{d y}{d x} = \frac{x^{5}}{y}$ by $4 + x^{6}$ .

$\frac{(4 + x^{6}) \frac{d y}{d x}}{4 + x^{6}} = \frac{\frac{x^{5}}{y}}{4 + x^{6}}$

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of $4 + x^{6}$ .

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Step 1.1.2.1.1

Cancel the common factor.

$\frac{(4 + x^{6}) \frac{d y}{d x}}{4 + x^{6}} = \frac{\frac{x^{5}}{y}}{4 + x^{6}}$

Step 1.1.2.1.2

Divide $\frac{d y}{d x}$ by $1$ .

$\frac{d y}{d x} = \frac{\frac{x^{5}}{y}}{4 + x^{6}}$

Step 1.1.3

Simplify the right side.

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Step 1.1.3.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{d y}{d x} = \frac{x^{5}}{y} \cdot \frac{1}{4 + x^{6}}$

Step 1.1.3.2

Multiply $\frac{x^{5}}{y}$ by $\frac{1}{4 + x^{6}}$ .

$\frac{d y}{d x} = \frac{x^{5}}{y (4 + x^{6})}$

Step 1.2

Regroup factors.

$\frac{d y}{d x} = \frac{x^{5}}{4 + x^{6}} \cdot \frac{1}{y}$

Step 1.3

Multiply both sides by $y$ .

$y \frac{d y}{d x} = y (\frac{x^{5}}{4 + x^{6}} \cdot \frac{1}{y})$

Step 1.4

Simplify.

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Step 1.4.1

Multiply $\frac{x^{5}}{4 + x^{6}}$ by $\frac{1}{y}$ .

$y \frac{d y}{d x} = y \frac{x^{5}}{(4 + x^{6}) y}$

Step 1.4.2

Cancel the common factor of $y$ .

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Step 1.4.2.1

Factor $y$ out of $(4 + x^{6}) y$ .

$y \frac{d y}{d x} = y \frac{x^{5}}{y (4 + x^{6})}$

Step 1.4.2.2

Cancel the common factor.

$y \frac{d y}{d x} = y \frac{x^{5}}{y (4 + x^{6})}$

Step 1.4.2.3

Rewrite the expression.

$y \frac{d y}{d x} = \frac{x^{5}}{4 + x^{6}}$

Step 1.5

Rewrite the equation.

$y d y = \frac{x^{5}}{4 + x^{6}} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y d y = \int \frac{x^{5}}{4 + x^{6}} d x$

Step 2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Rewrite $x^{6}$ as ${(x^{6})}^{1}$ .

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

Step 2.3.2

Let $u_{1} = x^{6}$ . Then $d u_{1} = 6 x^{5} d x$ , so $\frac{1}{6} d u_{1} = x^{5} d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 2.3.2.1

Let $u_{1} = x^{6}$ . Find $\frac{d u_{1}}{d x}$ .

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Step 2.3.2.1.1

Differentiate $x^{6}$ .

$\frac{d}{d x} [x^{6}]$

Step 2.3.2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$6 x^{5}$

Step 2.3.2.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\frac{1}{2} y^{2} + C_{1} = \int \frac{1}{4 + {u_{1}}^{1}} \cdot \frac{1}{6} d u_{1}$

Step 2.3.3

Simplify.

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Step 2.3.3.1

Simplify.

$\frac{1}{2} y^{2} + C_{1} = \int \frac{1}{4 + u_{1}} \cdot \frac{1}{6} d u_{1}$

Step 2.3.3.2

Multiply $\frac{1}{4 + u_{1}}$ by $\frac{1}{6}$ .

$\frac{1}{2} y^{2} + C_{1} = \int \frac{1}{(4 + u_{1}) \cdot 6} d u_{1}$

Step 2.3.3.3

Move $6$ to the left of $4 + u_{1}$ .

$\frac{1}{2} y^{2} + C_{1} = \int \frac{1}{6 (4 + u_{1})} d u_{1}$

Step 2.3.4

Since $\frac{1}{6}$ is constant with respect to $u_{1}$ , move $\frac{1}{6}$ out of the integral.

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{6} \int \frac{1}{4 + u_{1}} d u_{1}$

Step 2.3.5

Let $u_{2} = 4 + u_{1}$ . Then $d u_{2} = d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 2.3.5.1

Let $u_{2} = 4 + u_{1}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 2.3.5.1.1

Differentiate $4 + u_{1}$ .

$\frac{d}{d u_{1}} [4 + u_{1}]$

Step 2.3.5.1.2

By the Sum Rule, the derivative of $4 + u_{1}$ with respect to $u_{1}$ is $\frac{d}{d u_{1}} [4] + \frac{d}{d u_{1}} [u_{1}]$ .

$\frac{d}{d u_{1}} [4] + \frac{d}{d u_{1}} [u_{1}]$

Step 2.3.5.1.3

Since $4$ is constant with respect to $u_{1}$ , the derivative of $4$ with respect to $u_{1}$ is $0$ .

$0 + \frac{d}{d u_{1}} [u_{1}]$

Step 2.3.5.1.4

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 1$ .

$0 + 1$

Step 2.3.5.1.5

Add $0$ and $1$ .

$1$

Step 2.3.5.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{6} \int \frac{1}{u_{2}} d u_{2}$

Step 2.3.6

The integral of $\frac{1}{u_{2}}$ with respect to $u_{2}$ is $\ln (| u_{2} |)$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{6} (\ln (| u_{2} |) + C_{2})$

Step 2.3.7

Simplify.

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{6} \ln (| u_{2} |) + C_{2}$

Step 2.3.8

Substitute back in for each integration substitution variable.

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Step 2.3.8.1

Replace all occurrences of $u_{2}$ with $4 + u_{1}$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{6} \ln (| 4 + u_{1} |) + C_{2}$

Step 2.3.8.2

Replace all occurrences of $u_{1}$ with $x^{6}$ .

$\frac{1}{2} y^{2} + C_{1} = \frac{1}{6} \ln (| 4 + x^{6} |) + C_{2}$

Step 2.4

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

$\frac{1}{2} y^{2} = \frac{1}{6} \ln (| 4 + x^{6} |) + C$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} y^{2}) = 2 (\frac{1}{6} \ln (| 4 + x^{6} |) + C)$

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 $y^{2}$ .

$2 \frac{y^{2}}{2} = 2 (\frac{1}{6} \ln (| 4 + x^{6} |) + C)$

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.

$2 \frac{y^{2}}{2} = 2 (\frac{1}{6} \ln (| 4 + x^{6} |) + C)$

Step 3.2.1.1.2.2

Rewrite the expression.

$y^{2} = 2 (\frac{1}{6} \ln (| 4 + x^{6} |) + C)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $2 (\frac{1}{6} \ln (| 4 + x^{6} |) + C)$ .

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Step 3.2.2.1.1

Combine $\frac{1}{6}$ and $\ln (| 4 + x^{6} |)$ .

$y^{2} = 2 (\frac{\ln (| 4 + x^{6} |)}{6} + C)$

Step 3.2.2.1.2

To write $C$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$y^{2} = 2 (\frac{\ln (| 4 + x^{6} |)}{6} + C \cdot \frac{6}{6})$

Step 3.2.2.1.3

Simplify terms.

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Step 3.2.2.1.3.1

Combine $C$ and $\frac{6}{6}$ .

$y^{2} = 2 (\frac{\ln (| 4 + x^{6} |)}{6} + \frac{C \cdot 6}{6})$

Step 3.2.2.1.3.2

Combine the numerators over the common denominator.

$y^{2} = 2 \frac{\ln (| 4 + x^{6} |) + C \cdot 6}{6}$

Step 3.2.2.1.3.3

Cancel the common factor of $2$ .

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Step 3.2.2.1.3.3.1

Factor $2$ out of $6$ .

$y^{2} = 2 \frac{\ln (| 4 + x^{6} |) + C \cdot 6}{2 (3)}$

Step 3.2.2.1.3.3.2

Cancel the common factor.

$y^{2} = 2 \frac{\ln (| 4 + x^{6} |) + C \cdot 6}{2 \cdot 3}$

Step 3.2.2.1.3.3.3

Rewrite the expression.

$y^{2} = \frac{\ln (| 4 + x^{6} |) + C \cdot 6}{3}$

Step 3.2.2.1.4

Move $6$ to the left of $C$ .

$y^{2} = \frac{\ln (| 4 + x^{6} |) + 6 C}{3}$

Step 3.3

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

$y = \pm \sqrt{\frac{\ln (| 4 + x^{6} |) + 6 C}{3}}$

Step 3.4

Simplify $\pm \sqrt{\frac{\ln (| 4 + x^{6} |) + 6 C}{3}}$ .

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Step 3.4.1

Rewrite $\sqrt{\frac{\ln (| 4 + x^{6} |) + 6 C}{3}}$ as $\frac{\sqrt{\ln (| 4 + x^{6} |) + 6 C}}{\sqrt{3}}$ .

$y = \pm \frac{\sqrt{\ln (| 4 + x^{6} |) + 6 C}}{\sqrt{3}}$

Step 3.4.2

Multiply $\frac{\sqrt{\ln (| 4 + x^{6} |) + 6 C}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = \pm \frac{\sqrt{\ln (| 4 + x^{6} |) + 6 C}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 3.4.3

Combine and simplify the denominator.

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Step 3.4.3.1

Multiply $\frac{\sqrt{\ln (| 4 + x^{6} |) + 6 C}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$y = \pm \frac{\sqrt{\ln (| 4 + x^{6} |) + 6 C} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 3.4.3.2

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

$y = \pm \frac{\sqrt{\ln (| 4 + x^{6} |) + 6 C} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 3.4.3.3

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

$y = \pm \frac{\sqrt{\ln (| 4 + x^{6} |) + 6 C} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 3.4.3.4

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

$y = \pm \frac{\sqrt{\ln (| 4 + x^{6} |) + 6 C} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 3.4.3.5

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{\ln (| 4 + x^{6} |) + 6 C} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 3.4.3.6

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

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Step 3.4.3.6.1

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

$y = \pm \frac{\sqrt{\ln (| 4 + x^{6} |) + 6 C} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 3.4.3.6.2

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

$y = \pm \frac{\sqrt{\ln (| 4 + x^{6} |) + 6 C} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 3.4.3.6.3

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

$y = \pm \frac{\sqrt{\ln (| 4 + x^{6} |) + 6 C} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.4.3.6.4

Cancel the common factor of $2$ .

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Step 3.4.3.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{\ln (| 4 + x^{6} |) + 6 C} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 3.4.3.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{\ln (| 4 + x^{6} |) + 6 C} \sqrt{3}}{3^{1}}$

Step 3.4.3.6.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{\ln (| 4 + x^{6} |) + 6 C} \sqrt{3}}{3}$

Step 3.4.4

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{(\ln (| 4 + x^{6} |) + 6 C) \cdot 3}}{3}$

Step 3.4.5

Reorder factors in $\pm \frac{\sqrt{(\ln (| 4 + x^{6} |) + 6 C) \cdot 3}}{3}$ .

$y = \pm \frac{\sqrt{3 (\ln (| 4 + x^{6} |) + 6 C)}}{3}$

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.

$y = \frac{\sqrt{3 (\ln (| 4 + x^{6} |) + 6 C)}}{3}$

Step 3.5.2

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

$y = - \frac{\sqrt{3 (\ln (| 4 + x^{6} |) + 6 C)}}{3}$

Step 3.5.3

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

$y = \frac{\sqrt{3 (\ln (| 4 + x^{6} |) + 6 C)}}{3}$

$y = - \frac{\sqrt{3 (\ln (| 4 + x^{6} |) + 6 C)}}{3}$

$y = \frac{\sqrt{3 (\ln (| 4 + x^{6} |) + 6 C)}}{3}$

$y = - \frac{\sqrt{3 (\ln (| 4 + x^{6} |) + 6 C)}}{3}$

$y = \frac{\sqrt{3 (\ln (| 4 + x^{6} |) + 6 C)}}{3}$

$y = - \frac{\sqrt{3 (\ln (| 4 + x^{6} |) + 6 C)}}{3}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt{3 (\ln (| 4 + x^{6} |) + C)}}{3}$

$y = - \frac{\sqrt{3 (\ln (| 4 + x^{6} |) + C)}}{3}$