Calculus Examples

$\frac{d y}{d x} = 3 x^{5} \sqrt{y}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{\sqrt{y}}$ .

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{1}{\sqrt{y}} (3 x^{5} \sqrt{y})$

Step 1.2

Simplify.

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Step 1.2.1

Rewrite using the commutative property of multiplication.

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 3 \frac{1}{\sqrt{y}} (x^{5} \sqrt{y})$

Step 1.2.2

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 3 (\frac{1}{\sqrt{y}} \cdot \frac{\sqrt{y}}{\sqrt{y}}) (x^{5} \sqrt{y})$

Step 1.2.3

Combine and simplify the denominator.

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Step 1.2.3.1

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 3 \frac{\sqrt{y}}{\sqrt{y} \sqrt{y}} (x^{5} \sqrt{y})$

Step 1.2.3.2

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 3 \frac{\sqrt{y}}{{\sqrt{y}}^{1} \sqrt{y}} (x^{5} \sqrt{y})$

Step 1.2.3.3

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 3 \frac{\sqrt{y}}{{\sqrt{y}}^{1} {\sqrt{y}}^{1}} (x^{5} \sqrt{y})$

Step 1.2.3.4

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 3 \frac{\sqrt{y}}{{\sqrt{y}}^{1 + 1}} (x^{5} \sqrt{y})$

Step 1.2.3.5

Add $1$ and $1$ .

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 3 \frac{\sqrt{y}}{{\sqrt{y}}^{2}} (x^{5} \sqrt{y})$

Step 1.2.3.6

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

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Step 1.2.3.6.1

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 3 \frac{\sqrt{y}}{{(y^{\frac{1}{2}})}^{2}} (x^{5} \sqrt{y})$

Step 1.2.3.6.2

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 3 \frac{\sqrt{y}}{y^{\frac{1}{2} \cdot 2}} (x^{5} \sqrt{y})$

Step 1.2.3.6.3

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 3 \frac{\sqrt{y}}{y^{\frac{2}{2}}} (x^{5} \sqrt{y})$

Step 1.2.3.6.4

Cancel the common factor of $2$ .

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Step 1.2.3.6.4.1

Cancel the common factor.

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 3 \frac{\sqrt{y}}{y^{\frac{2}{2}}} (x^{5} \sqrt{y})$

Step 1.2.3.6.4.2

Rewrite the expression.

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 3 \frac{\sqrt{y}}{y^{1}} (x^{5} \sqrt{y})$

Step 1.2.3.6.5

Simplify.

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 3 \frac{\sqrt{y}}{y} (x^{5} \sqrt{y})$

Step 1.2.4

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{3 \sqrt{y}}{y} (x^{5} \sqrt{y})$

Step 1.2.5

Multiply $\frac{3 \sqrt{y}}{y} (x^{5} \sqrt{y})$ .

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Step 1.2.5.1

Combine $x^{5}$ and $\frac{3 \sqrt{y}}{y}$ .

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{x^{5} (3 \sqrt{y})}{y} \sqrt{y}$

Step 1.2.5.2

Combine $\frac{x^{5} (3 \sqrt{y})}{y}$ and $\sqrt{y}$ .

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{x^{5} (3 \sqrt{y}) \sqrt{y}}{y}$

Step 1.2.5.3

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{x^{5} (3 ({\sqrt{y}}^{1} \sqrt{y}))}{y}$

Step 1.2.5.4

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{x^{5} (3 ({\sqrt{y}}^{1} {\sqrt{y}}^{1}))}{y}$

Step 1.2.5.5

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{x^{5} (3 {\sqrt{y}}^{1 + 1})}{y}$

Step 1.2.5.6

Add $1$ and $1$ .

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{x^{5} (3 {\sqrt{y}}^{2})}{y}$

Step 1.2.6

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

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Step 1.2.6.1

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{x^{5} \cdot 3 {(y^{\frac{1}{2}})}^{2}}{y}$

Step 1.2.6.2

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{x^{5} \cdot 3 y^{\frac{1}{2} \cdot 2}}{y}$

Step 1.2.6.3

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

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{x^{5} \cdot 3 y^{\frac{2}{2}}}{y}$

Step 1.2.6.4

Cancel the common factor of $2$ .

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Step 1.2.6.4.1

Cancel the common factor.

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{x^{5} \cdot 3 y^{\frac{2}{2}}}{y}$

Step 1.2.6.4.2

Rewrite the expression.

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{x^{5} \cdot 3 y^{1}}{y}$

Step 1.2.6.5

Simplify.

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{x^{5} \cdot 3 y}{y}$

Step 1.2.7

Cancel the common factor of $y$ .

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Step 1.2.7.1

Cancel the common factor.

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = \frac{x^{5} \cdot 3 y}{y}$

Step 1.2.7.2

Divide $x^{5} \cdot 3$ by $1$ .

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = x^{5} \cdot 3$

Step 1.2.8

Move $3$ to the left of $x^{5}$ .

$\frac{1}{\sqrt{y}} \frac{d y}{d x} = 3 x^{5}$

Step 1.3

Rewrite the equation.

$\frac{1}{\sqrt{y}} d y = 3 x^{5} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{\sqrt{y}} d y = \int 3 x^{5} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Apply basic rules of exponents.

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Step 2.2.1.1

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

$\int \frac{1}{y^{\frac{1}{2}}} d y = \int 3 x^{5} d x$

Step 2.2.1.2

Move $y^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

$\int {(y^{\frac{1}{2}})}^{- 1} d y = \int 3 x^{5} d x$

Step 2.2.1.3

Multiply the exponents in ${(y^{\frac{1}{2}})}^{- 1}$ .

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Step 2.2.1.3.1

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

$\int y^{\frac{1}{2} \cdot - 1} d y = \int 3 x^{5} d x$

Step 2.2.1.3.2

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

$\int y^{\frac{- 1}{2}} d y = \int 3 x^{5} d x$

Step 2.2.1.3.3

Move the negative in front of the fraction.

$\int y^{- \frac{1}{2}} d y = \int 3 x^{5} d x$

Step 2.2.2

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

$2 y^{\frac{1}{2}} + C_{1} = \int 3 x^{5} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$2 y^{\frac{1}{2}} + C_{1} = 3 \int x^{5} d x$

Step 2.3.2

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

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

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $3 (\frac{1}{6} x^{6} + C_{2})$ as $3 (\frac{1}{6}) x^{6} + C_{2}$ .

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

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

Combine $3$ and $\frac{1}{6}$ .

$2 y^{\frac{1}{2}} + C_{1} = \frac{3}{6} x^{6} + C_{2}$

Step 2.3.3.2.2

Cancel the common factor of $3$ and $6$ .

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Step 2.3.3.2.2.1

Factor $3$ out of $3$ .

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

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 $6$ .

$2 y^{\frac{1}{2}} + C_{1} = \frac{3 \cdot 1}{3 \cdot 2} x^{6} + C_{2}$

Step 2.3.3.2.2.2.2

Cancel the common factor.

$2 y^{\frac{1}{2}} + C_{1} = \frac{3 \cdot 1}{3 \cdot 2} x^{6} + C_{2}$

Step 2.3.3.2.2.2.3

Rewrite the expression.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

Divide each term in $2 y^{\frac{1}{2}} = \frac{1}{2} x^{6} + K$ by $2$ and simplify.

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Step 3.1.1

Divide each term in $2 y^{\frac{1}{2}} = \frac{1}{2} x^{6} + K$ by $2$ .

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

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor.

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

Step 3.1.2.2

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

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

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 $x^{6}$ .

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

Step 3.1.3.1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.1.3.1.3

Combine.

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

Step 3.1.3.1.4

Multiply $x^{6}$ by $1$ .

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

Step 3.1.3.1.5

Multiply $2$ by $2$ .

$y^{\frac{1}{2}} = \frac{x^{6}}{4} + \frac{K}{2}$

Step 3.2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Step 3.3

Simplify the exponent.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

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

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Step 3.3.1.1.1

Multiply the exponents in ${(y^{\frac{1}{2}})}^{2}$ .

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Step 3.3.1.1.1.1

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

$y^{\frac{1}{2} \cdot 2} = {(\frac{x^{6}}{4} + \frac{K}{2})}^{2}$

Step 3.3.1.1.1.2

Cancel the common factor of $2$ .

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Step 3.3.1.1.1.2.1

Cancel the common factor.

$y^{\frac{1}{2} \cdot 2} = {(\frac{x^{6}}{4} + \frac{K}{2})}^{2}$

Step 3.3.1.1.1.2.2

Rewrite the expression.

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

Step 3.3.1.1.2

Simplify.

$y = {(\frac{x^{6}}{4} + \frac{K}{2})}^{2}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify ${(\frac{x^{6}}{4} + \frac{K}{2})}^{2}$ .

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Step 3.3.2.1.1

Rewrite ${(\frac{x^{6}}{4} + \frac{K}{2})}^{2}$ as $(\frac{x^{6}}{4} + \frac{K}{2}) (\frac{x^{6}}{4} + \frac{K}{2})$ .

$y = (\frac{x^{6}}{4} + \frac{K}{2}) (\frac{x^{6}}{4} + \frac{K}{2})$

Step 3.3.2.1.2

Expand $(\frac{x^{6}}{4} + \frac{K}{2}) (\frac{x^{6}}{4} + \frac{K}{2})$ using the FOIL Method.

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Step 3.3.2.1.2.1

Apply the distributive property.

$y = \frac{x^{6}}{4} (\frac{x^{6}}{4} + \frac{K}{2}) + \frac{K}{2} (\frac{x^{6}}{4} + \frac{K}{2})$

Step 3.3.2.1.2.2

Apply the distributive property.

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

Step 3.3.2.1.2.3

Apply the distributive property.

$y = \frac{x^{6}}{4} \cdot \frac{x^{6}}{4} + \frac{x^{6}}{4} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{6}}{4} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3

Simplify and combine like terms.

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Step 3.3.2.1.3.1

Simplify each term.

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Step 3.3.2.1.3.1.1

Combine.

$y = \frac{x^{6} x^{6}}{4 \cdot 4} + \frac{x^{6}}{4} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{6}}{4} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.2

Multiply $x^{6}$ by $x^{6}$ by adding the exponents.

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Step 3.3.2.1.3.1.2.1

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

$y = \frac{x^{6 + 6}}{4 \cdot 4} + \frac{x^{6}}{4} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{6}}{4} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.2.2

Add $6$ and $6$ .

$y = \frac{x^{12}}{4 \cdot 4} + \frac{x^{6}}{4} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{6}}{4} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.3

Multiply $4$ by $4$ .

$y = \frac{x^{12}}{16} + \frac{x^{6}}{4} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{6}}{4} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.4

Combine.

$y = \frac{x^{12}}{16} + \frac{x^{6} K}{4 \cdot 2} + \frac{K}{2} \cdot \frac{x^{6}}{4} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.5

Multiply $4$ by $2$ .

$y = \frac{x^{12}}{16} + \frac{x^{6} K}{8} + \frac{K}{2} \cdot \frac{x^{6}}{4} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.6

Combine.

$y = \frac{x^{12}}{16} + \frac{x^{6} K}{8} + \frac{K x^{6}}{2 \cdot 4} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.7

Multiply $2$ by $4$ .

$y = \frac{x^{12}}{16} + \frac{x^{6} K}{8} + \frac{K x^{6}}{8} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.8

Multiply $\frac{K}{2} \cdot \frac{K}{2}$ .

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Step 3.3.2.1.3.1.8.1

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

$y = \frac{x^{12}}{16} + \frac{x^{6} K}{8} + \frac{K x^{6}}{8} + \frac{K \cdot K}{2 \cdot 2}$

Step 3.3.2.1.3.1.8.2

Raise $K$ to the power of $1$ .

$y = \frac{x^{12}}{16} + \frac{x^{6} K}{8} + \frac{K x^{6}}{8} + \frac{K^{1} K}{2 \cdot 2}$

Step 3.3.2.1.3.1.8.3

Raise $K$ to the power of $1$ .

$y = \frac{x^{12}}{16} + \frac{x^{6} K}{8} + \frac{K x^{6}}{8} + \frac{K^{1} K^{1}}{2 \cdot 2}$

Step 3.3.2.1.3.1.8.4

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

$y = \frac{x^{12}}{16} + \frac{x^{6} K}{8} + \frac{K x^{6}}{8} + \frac{K^{1 + 1}}{2 \cdot 2}$

Step 3.3.2.1.3.1.8.5

Add $1$ and $1$ .

$y = \frac{x^{12}}{16} + \frac{x^{6} K}{8} + \frac{K x^{6}}{8} + \frac{K^{2}}{2 \cdot 2}$

Step 3.3.2.1.3.1.8.6

Multiply $2$ by $2$ .

$y = \frac{x^{12}}{16} + \frac{x^{6} K}{8} + \frac{K x^{6}}{8} + \frac{K^{2}}{4}$

Step 3.3.2.1.3.2

Add $\frac{x^{6} K}{8}$ and $\frac{K x^{6}}{8}$ .

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Step 3.3.2.1.3.2.1

Reorder $x^{6}$ and $K$ .

$y = \frac{x^{12}}{16} + \frac{K x^{6}}{8} + \frac{K x^{6}}{8} + \frac{K^{2}}{4}$

Step 3.3.2.1.3.2.2

Add $\frac{K x^{6}}{8}$ and $\frac{K x^{6}}{8}$ .

$y = \frac{x^{12}}{16} + 2 \frac{K x^{6}}{8} + \frac{K^{2}}{4}$

Step 3.3.2.1.4

Cancel the common factor of $2$ .

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Step 3.3.2.1.4.1

Factor $2$ out of $8$ .

$y = \frac{x^{12}}{16} + 2 \frac{K x^{6}}{2 (4)} + \frac{K^{2}}{4}$

Step 3.3.2.1.4.2

Cancel the common factor.

$y = \frac{x^{12}}{16} + 2 \frac{K x^{6}}{2 \cdot 4} + \frac{K^{2}}{4}$

Step 3.3.2.1.4.3

Rewrite the expression.

$y = \frac{x^{12}}{16} + \frac{K x^{6}}{4} + \frac{K^{2}}{4}$

Step 4

Simplify the constant of integration.

$y = \frac{x^{12}}{16} + \frac{K x^{6}}{4} + K$