Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{3 + y^{2}}$ .

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

Step 1.2

Cancel the common factor of $3 + y^{2}$ .

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Step 1.2.1

Factor $3 + y^{2}$ out of $(2 + x^{2}) (3 + y^{2})$ .

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

Step 1.2.2

Cancel the common factor.

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

Step 1.2.3

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{3 + y^{2}} d y = \int 2 + x^{2} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

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

$\int \frac{1}{{\sqrt{3}}^{2} + y^{2}} d y = \int 2 + x^{2} d x$

Step 2.2.2

The integral of $\frac{1}{{\sqrt{3}}^{2} + y^{2}}$ with respect to $y$ is $\frac{1}{\sqrt{3}} \arctan (\frac{y}{\sqrt{3}}) + C_{1}$ .

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

Step 2.2.3

Simplify the answer.

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Step 2.2.3.1

Combine $\frac{1}{\sqrt{3}}$ and $\arctan (\frac{y}{\sqrt{3}})$ .

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

Step 2.2.3.2

Rewrite $\frac{\arctan (\frac{y}{\sqrt{3}})}{\sqrt{3}} + C_{1}$ as $\frac{\arctan (\frac{y \sqrt{3}}{3})}{\sqrt{3}} + C_{1}$ .

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

Step 2.2.3.3

Reorder terms.

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

Apply the constant rule.

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

Step 2.3.3

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

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

Step 2.3.4

Simplify.

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

Step 2.3.5

Reorder terms.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Simplify.

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Step 3.1.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 3.1.2

Combine and simplify the denominator.

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Step 3.1.2.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 3.1.2.2

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

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

Step 3.1.2.3

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

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

Step 3.1.2.4

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

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

Step 3.1.2.5

Add $1$ and $1$ .

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

Step 3.1.2.6

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

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Step 3.1.2.6.1

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

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

Step 3.1.2.6.2

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

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

Step 3.1.2.6.3

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

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

Step 3.1.2.6.4

Cancel the common factor of $2$ .

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Step 3.1.2.6.4.1

Cancel the common factor.

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

Step 3.1.2.6.4.2

Rewrite the expression.

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

Step 3.1.2.6.5

Evaluate the exponent.

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

Step 3.2

Multiply each term in $\frac{\sqrt{3}}{3} \cdot \arctan (\frac{\sqrt{3}}{3} y) = \frac{1}{3} x^{3} + 2 x + K$ by $\frac{3}{\sqrt{3}}$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $\frac{\sqrt{3}}{3} \cdot \arctan (\frac{\sqrt{3}}{3} y) = \frac{1}{3} x^{3} + 2 x + K$ by $\frac{3}{\sqrt{3}}$ .

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Combine $\frac{\sqrt{3}}{3}$ and $y$ .

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

Step 3.2.2.2

Combine $\frac{\sqrt{3}}{3}$ and $\arctan (\frac{\sqrt{3} y}{3})$ .

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

Step 3.2.2.3

Cancel the common factor of $\sqrt{3}$ .

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Step 3.2.2.3.1

Factor $\sqrt{3}$ out of $\sqrt{3} \arctan (\frac{\sqrt{3} y}{3})$ .

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

Step 3.2.2.3.2

Cancel the common factor.

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

Step 3.2.2.3.3

Rewrite the expression.

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

Step 3.2.2.4

Cancel the common factor of $3$ .

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Step 3.2.2.4.1

Cancel the common factor.

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

Step 3.2.2.4.2

Rewrite the expression.

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

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

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

Step 3.2.3.1.2

Combine.

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

Step 3.2.3.1.3

Cancel the common factor of $3$ .

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Step 3.2.3.1.3.1

Cancel the common factor.

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

Step 3.2.3.1.3.2

Rewrite the expression.

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

Step 3.2.3.1.4

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

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

Step 3.2.3.1.5

Combine and simplify the denominator.

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Step 3.2.3.1.5.1

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

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

Step 3.2.3.1.5.2

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

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

Step 3.2.3.1.5.3

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

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

Step 3.2.3.1.5.4

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

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

Step 3.2.3.1.5.5

Add $1$ and $1$ .

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

Step 3.2.3.1.5.6

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

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Step 3.2.3.1.5.6.1

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

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

Step 3.2.3.1.5.6.2

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

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

Step 3.2.3.1.5.6.3

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

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

Step 3.2.3.1.5.6.4

Cancel the common factor of $2$ .

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Step 3.2.3.1.5.6.4.1

Cancel the common factor.

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

Step 3.2.3.1.5.6.4.2

Rewrite the expression.

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

Step 3.2.3.1.5.6.5

Evaluate the exponent.

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

Step 3.2.3.1.6

Multiply $\frac{3}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 3.2.3.1.7

Combine and simplify the denominator.

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Step 3.2.3.1.7.1

Multiply $\frac{3}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 3.2.3.1.7.2

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

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

Step 3.2.3.1.7.3

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

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

Step 3.2.3.1.7.4

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

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

Step 3.2.3.1.7.5

Add $1$ and $1$ .

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

Step 3.2.3.1.7.6

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

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Step 3.2.3.1.7.6.1

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

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

Step 3.2.3.1.7.6.2

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

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

Step 3.2.3.1.7.6.3

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

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

Step 3.2.3.1.7.6.4

Cancel the common factor of $2$ .

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Step 3.2.3.1.7.6.4.1

Cancel the common factor.

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

Step 3.2.3.1.7.6.4.2

Rewrite the expression.

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

Step 3.2.3.1.7.6.5

Evaluate the exponent.

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

Step 3.2.3.1.8

Cancel the common factor of $3$ .

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Step 3.2.3.1.8.1

Cancel the common factor.

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

Step 3.2.3.1.8.2

Divide $\sqrt{3}$ by $1$ .

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

Step 3.2.3.1.9

Multiply $\frac{3}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 3.2.3.1.10

Combine and simplify the denominator.

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Step 3.2.3.1.10.1

Multiply $\frac{3}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 3.2.3.1.10.2

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

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

Step 3.2.3.1.10.3

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

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

Step 3.2.3.1.10.4

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

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

Step 3.2.3.1.10.5

Add $1$ and $1$ .

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

Step 3.2.3.1.10.6

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

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Step 3.2.3.1.10.6.1

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

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

Step 3.2.3.1.10.6.2

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

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

Step 3.2.3.1.10.6.3

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

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

Step 3.2.3.1.10.6.4

Cancel the common factor of $2$ .

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Step 3.2.3.1.10.6.4.1

Cancel the common factor.

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

Step 3.2.3.1.10.6.4.2

Rewrite the expression.

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

Step 3.2.3.1.10.6.5

Evaluate the exponent.

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

Step 3.2.3.1.11

Cancel the common factor of $3$ .

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Step 3.2.3.1.11.1

Cancel the common factor.

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

Step 3.2.3.1.11.2

Divide $\sqrt{3}$ by $1$ .

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

Step 3.3

Take the inverse arctangent of both sides of the equation to extract $y$ from inside the arctangent.

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

Step 3.4

Multiply both sides of the equation by $\frac{3}{\sqrt{3}}$ .

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

Step 3.5

Simplify both sides of the equation.

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Step 3.5.1

Simplify the left side.

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Step 3.5.1.1

Simplify $\frac{3}{\sqrt{3}} \cdot \frac{\sqrt{3} y}{3}$ .

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Step 3.5.1.1.1

Cancel the common factor of $3$ .

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Step 3.5.1.1.1.1

Cancel the common factor.

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

Step 3.5.1.1.1.2

Rewrite the expression.

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

Step 3.5.1.1.2

Cancel the common factor of $\sqrt{3}$ .

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Step 3.5.1.1.2.1

Factor $\sqrt{3}$ out of $\sqrt{3} y$ .

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

Step 3.5.1.1.2.2

Cancel the common factor.

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

Step 3.5.1.1.2.3

Rewrite the expression.

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

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

Simplify $\frac{3}{\sqrt{3}} \tan (\frac{x^{3} \sqrt{3}}{3} + 2 x \sqrt{3} + K \sqrt{3})$ .

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Step 3.5.2.1.1

Multiply $\frac{3}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 3.5.2.1.2

Combine and simplify the denominator.

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Step 3.5.2.1.2.1

Multiply $\frac{3}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 3.5.2.1.2.2

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

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

Step 3.5.2.1.2.3

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

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

Step 3.5.2.1.2.4

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

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

Step 3.5.2.1.2.5

Add $1$ and $1$ .

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

Step 3.5.2.1.2.6

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

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Step 3.5.2.1.2.6.1

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

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

Step 3.5.2.1.2.6.2

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

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

Step 3.5.2.1.2.6.3

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

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

Step 3.5.2.1.2.6.4

Cancel the common factor of $2$ .

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Step 3.5.2.1.2.6.4.1

Cancel the common factor.

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

Step 3.5.2.1.2.6.4.2

Rewrite the expression.

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

Step 3.5.2.1.2.6.5

Evaluate the exponent.

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

Step 3.5.2.1.3

Cancel the common factor of $3$ .

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Step 3.5.2.1.3.1

Cancel the common factor.

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

Step 3.5.2.1.3.2

Divide $\sqrt{3}$ by $1$ .

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

Step 4

Simplify the constant of integration.

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