Calculus Examples

$\frac{d y}{d x} = (1 + 2 x) \sqrt{y}$ , $y (0) = 1$

Step 1

Separate the variables.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Factor $\sqrt{y}$ out of $(1 + 2 x) \sqrt{y}$ .

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

Step 1.2.2

Cancel the common factor.

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

Step 1.2.3

Rewrite the expression.

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

Step 1.3

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

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

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

Step 2.2.1.2

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

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.2.1.3.2

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

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

Step 2.2.1.3.3

Move the negative in front of the fraction.

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

Step 2.2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Split the single integral into multiple integrals.

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

Step 2.3.2

Apply the constant rule.

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

Step 2.3.3

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 2.3.4

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

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

Step 2.3.5

Simplify.

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Step 2.3.5.1

Simplify.

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

Step 2.3.5.2

Simplify.

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Step 2.3.5.2.1

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

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

Step 2.3.5.2.2

Cancel the common factor of $2$ .

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Step 2.3.5.2.2.1

Cancel the common factor.

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

Step 2.3.5.2.2.2

Rewrite the expression.

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

Step 2.3.5.2.3

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

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

Step 2.3.6

Reorder terms.

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

Step 2.4

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

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

Step 3

Solve for $y$ .

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor.

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

Step 3.1.2.2

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

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

Step 3.2

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

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

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.

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

Step 3.3.1.1.1.2.2

Rewrite the expression.

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

Step 3.3.1.1.2

Simplify.

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

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

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

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Step 3.3.2.1.1

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

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

Step 3.3.2.1.2

Expand $(\frac{x^{2}}{2} + \frac{x}{2} + \frac{K}{2}) (\frac{x^{2}}{2} + \frac{x}{2} + \frac{K}{2})$ by multiplying each term in the first expression by each term in the second expression.

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

Step 3.3.2.1.3

Simplify 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.

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

Step 3.3.2.1.3.1.2

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

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

Step 3.3.2.1.3.1.2.2

Add $2$ and $2$ .

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

Step 3.3.2.1.3.1.3

Multiply $2$ by $2$ .

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

Step 3.3.2.1.3.1.4

Combine.

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

Step 3.3.2.1.3.1.5

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 3.3.2.1.3.1.5.1

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

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Step 3.3.2.1.3.1.5.1.1

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

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

Step 3.3.2.1.3.1.5.1.2

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

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

Step 3.3.2.1.3.1.5.2

Add $2$ and $1$ .

$y = \frac{x^{4}}{4} + \frac{x^{3}}{2 \cdot 2} + \frac{x^{2}}{2} \cdot \frac{K}{2} + \frac{x}{2} \cdot \frac{x^{2}}{2} + \frac{x}{2} \cdot \frac{x}{2} + \frac{x}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{2}}{2} + \frac{K}{2} \cdot \frac{x}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.6

Multiply $2$ by $2$ .

$y = \frac{x^{4}}{4} + \frac{x^{3}}{4} + \frac{x^{2}}{2} \cdot \frac{K}{2} + \frac{x}{2} \cdot \frac{x^{2}}{2} + \frac{x}{2} \cdot \frac{x}{2} + \frac{x}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{2}}{2} + \frac{K}{2} \cdot \frac{x}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.7

Combine.

$y = \frac{x^{4}}{4} + \frac{x^{3}}{4} + \frac{x^{2} K}{2 \cdot 2} + \frac{x}{2} \cdot \frac{x^{2}}{2} + \frac{x}{2} \cdot \frac{x}{2} + \frac{x}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{2}}{2} + \frac{K}{2} \cdot \frac{x}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.8

Multiply $2$ by $2$ .

$y = \frac{x^{4}}{4} + \frac{x^{3}}{4} + \frac{x^{2} K}{4} + \frac{x}{2} \cdot \frac{x^{2}}{2} + \frac{x}{2} \cdot \frac{x}{2} + \frac{x}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{2}}{2} + \frac{K}{2} \cdot \frac{x}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.9

Combine.

$y = \frac{x^{4}}{4} + \frac{x^{3}}{4} + \frac{x^{2} K}{4} + \frac{x \cdot x^{2}}{2 \cdot 2} + \frac{x}{2} \cdot \frac{x}{2} + \frac{x}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{2}}{2} + \frac{K}{2} \cdot \frac{x}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.10

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 3.3.2.1.3.1.10.1

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

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Step 3.3.2.1.3.1.10.1.1

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

$y = \frac{x^{4}}{4} + \frac{x^{3}}{4} + \frac{x^{2} K}{4} + \frac{x^{1} x^{2}}{2 \cdot 2} + \frac{x}{2} \cdot \frac{x}{2} + \frac{x}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{2}}{2} + \frac{K}{2} \cdot \frac{x}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.10.1.2

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

$y = \frac{x^{4}}{4} + \frac{x^{3}}{4} + \frac{x^{2} K}{4} + \frac{x^{1 + 2}}{2 \cdot 2} + \frac{x}{2} \cdot \frac{x}{2} + \frac{x}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{2}}{2} + \frac{K}{2} \cdot \frac{x}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.10.2

Add $1$ and $2$ .

$y = \frac{x^{4}}{4} + \frac{x^{3}}{4} + \frac{x^{2} K}{4} + \frac{x^{3}}{2 \cdot 2} + \frac{x}{2} \cdot \frac{x}{2} + \frac{x}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{2}}{2} + \frac{K}{2} \cdot \frac{x}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.11

Multiply $2$ by $2$ .

$y = \frac{x^{4}}{4} + \frac{x^{3}}{4} + \frac{x^{2} K}{4} + \frac{x^{3}}{4} + \frac{x}{2} \cdot \frac{x}{2} + \frac{x}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{2}}{2} + \frac{K}{2} \cdot \frac{x}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.12

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

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Step 3.3.2.1.3.1.12.1

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

$y = \frac{x^{4}}{4} + \frac{x^{3}}{4} + \frac{x^{2} K}{4} + \frac{x^{3}}{4} + \frac{x \cdot x}{2 \cdot 2} + \frac{x}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{2}}{2} + \frac{K}{2} \cdot \frac{x}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.12.2

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

$y = \frac{x^{4}}{4} + \frac{x^{3}}{4} + \frac{x^{2} K}{4} + \frac{x^{3}}{4} + \frac{x^{1} x}{2 \cdot 2} + \frac{x}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{2}}{2} + \frac{K}{2} \cdot \frac{x}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.12.3

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

$y = \frac{x^{4}}{4} + \frac{x^{3}}{4} + \frac{x^{2} K}{4} + \frac{x^{3}}{4} + \frac{x^{1} x^{1}}{2 \cdot 2} + \frac{x}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{2}}{2} + \frac{K}{2} \cdot \frac{x}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.12.4

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

$y = \frac{x^{4}}{4} + \frac{x^{3}}{4} + \frac{x^{2} K}{4} + \frac{x^{3}}{4} + \frac{x^{1 + 1}}{2 \cdot 2} + \frac{x}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{2}}{2} + \frac{K}{2} \cdot \frac{x}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.12.5

Add $1$ and $1$ .

$y = \frac{x^{4}}{4} + \frac{x^{3}}{4} + \frac{x^{2} K}{4} + \frac{x^{3}}{4} + \frac{x^{2}}{2 \cdot 2} + \frac{x}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{2}}{2} + \frac{K}{2} \cdot \frac{x}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.12.6

Multiply $2$ by $2$ .

$y = \frac{x^{4}}{4} + \frac{x^{3}}{4} + \frac{x^{2} K}{4} + \frac{x^{3}}{4} + \frac{x^{2}}{4} + \frac{x}{2} \cdot \frac{K}{2} + \frac{K}{2} \cdot \frac{x^{2}}{2} + \frac{K}{2} \cdot \frac{x}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.13

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

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Step 3.3.2.1.3.1.13.1

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

$y = \frac{x^{4}}{4} + \frac{x^{3}}{4} + \frac{x^{2} K}{4} + \frac{x^{3}}{4} + \frac{x^{2}}{4} + \frac{x K}{2 \cdot 2} + \frac{K}{2} \cdot \frac{x^{2}}{2} + \frac{K}{2} \cdot \frac{x}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.13.2

Multiply $2$ by $2$ .

$y = \frac{x^{4}}{4} + \frac{x^{3}}{4} + \frac{x^{2} K}{4} + \frac{x^{3}}{4} + \frac{x^{2}}{4} + \frac{x K}{4} + \frac{K}{2} \cdot \frac{x^{2}}{2} + \frac{K}{2} \cdot \frac{x}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.14

Combine.

$y = \frac{x^{4}}{4} + \frac{x^{3}}{4} + \frac{x^{2} K}{4} + \frac{x^{3}}{4} + \frac{x^{2}}{4} + \frac{x K}{4} + \frac{K x^{2}}{2 \cdot 2} + \frac{K}{2} \cdot \frac{x}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.15

Multiply $2$ by $2$ .

$y = \frac{x^{4}}{4} + \frac{x^{3}}{4} + \frac{x^{2} K}{4} + \frac{x^{3}}{4} + \frac{x^{2}}{4} + \frac{x K}{4} + \frac{K x^{2}}{4} + \frac{K}{2} \cdot \frac{x}{2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.16

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

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Step 3.3.2.1.3.1.16.1

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

$y = \frac{x^{4}}{4} + \frac{x^{3}}{4} + \frac{x^{2} K}{4} + \frac{x^{3}}{4} + \frac{x^{2}}{4} + \frac{x K}{4} + \frac{K x^{2}}{4} + \frac{K x}{2 \cdot 2} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.16.2

Multiply $2$ by $2$ .

$y = \frac{x^{4}}{4} + \frac{x^{3}}{4} + \frac{x^{2} K}{4} + \frac{x^{3}}{4} + \frac{x^{2}}{4} + \frac{x K}{4} + \frac{K x^{2}}{4} + \frac{K x}{4} + \frac{K}{2} \cdot \frac{K}{2}$

Step 3.3.2.1.3.1.17

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

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Step 3.3.2.1.3.1.17.1

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

$y = \frac{x^{4}}{4} + \frac{x^{3}}{4} + \frac{x^{2} K}{4} + \frac{x^{3}}{4} + \frac{x^{2}}{4} + \frac{x K}{4} + \frac{K x^{2}}{4} + \frac{K x}{4} + \frac{K \cdot K}{2 \cdot 2}$

Step 3.3.2.1.3.1.17.2

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

$y = \frac{x^{4}}{4} + \frac{x^{3}}{4} + \frac{x^{2} K}{4} + \frac{x^{3}}{4} + \frac{x^{2}}{4} + \frac{x K}{4} + \frac{K x^{2}}{4} + \frac{K x}{4} + \frac{K^{1} K}{2 \cdot 2}$

Step 3.3.2.1.3.1.17.3

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

$y = \frac{x^{4}}{4} + \frac{x^{3}}{4} + \frac{x^{2} K}{4} + \frac{x^{3}}{4} + \frac{x^{2}}{4} + \frac{x K}{4} + \frac{K x^{2}}{4} + \frac{K x}{4} + \frac{K^{1} K^{1}}{2 \cdot 2}$

Step 3.3.2.1.3.1.17.4

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

$y = \frac{x^{4}}{4} + \frac{x^{3}}{4} + \frac{x^{2} K}{4} + \frac{x^{3}}{4} + \frac{x^{2}}{4} + \frac{x K}{4} + \frac{K x^{2}}{4} + \frac{K x}{4} + \frac{K^{1 + 1}}{2 \cdot 2}$

Step 3.3.2.1.3.1.17.5

Add $1$ and $1$ .

$y = \frac{x^{4}}{4} + \frac{x^{3}}{4} + \frac{x^{2} K}{4} + \frac{x^{3}}{4} + \frac{x^{2}}{4} + \frac{x K}{4} + \frac{K x^{2}}{4} + \frac{K x}{4} + \frac{K^{2}}{2 \cdot 2}$

Step 3.3.2.1.3.1.17.6

Multiply $2$ by $2$ .

$y = \frac{x^{4}}{4} + \frac{x^{3}}{4} + \frac{x^{2} K}{4} + \frac{x^{3}}{4} + \frac{x^{2}}{4} + \frac{x K}{4} + \frac{K x^{2}}{4} + \frac{K x}{4} + \frac{K^{2}}{4}$

Step 3.3.2.1.3.2

Simplify terms.

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Step 3.3.2.1.3.2.1

Combine the numerators over the common denominator.

$y = \frac{x^{4} + x^{3} + x^{2} K + x^{3} + x^{2} + x K + K x^{2} + K x + K^{2}}{4}$

Step 3.3.2.1.3.2.2

Add $x^{3}$ and $x^{3}$ .

$y = \frac{x^{4} + 2 x^{3} + x^{2} K + x^{2} + x K + K x^{2} + K x + K^{2}}{4}$

Step 3.3.2.1.4

Add $x^{2} K$ and $K x^{2}$ .

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Step 3.3.2.1.4.1

Reorder $x^{2}$ and $K$ .

$y = \frac{x^{4} + 2 x^{3} + x^{2} + x K + K x^{2} + K x^{2} + K x + K^{2}}{4}$

Step 3.3.2.1.4.2

Add $K x^{2}$ and $K x^{2}$ .

$y = \frac{x^{4} + 2 x^{3} + x^{2} + x K + 2 K x^{2} + K x + K^{2}}{4}$

Step 3.3.2.1.5

Add $x K$ and $K x$ .

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Step 3.3.2.1.5.1

Reorder $x$ and $K$ .

$y = \frac{x^{4} + 2 x^{3} + x^{2} + K x + K x + 2 K x^{2} + K^{2}}{4}$

Step 3.3.2.1.5.2

Add $K x$ and $K x$ .

$y = \frac{x^{4} + 2 x^{3} + x^{2} + 2 K x + 2 K x^{2} + K^{2}}{4}$

Step 3.3.2.1.6

Split the fraction $\frac{x^{4} + 2 x^{3} + x^{2} + 2 K x + 2 K x^{2} + K^{2}}{4}$ into two fractions.

$y = \frac{x^{4} + 2 x^{3} + x^{2} + 2 K x + 2 K x^{2}}{4} + \frac{K^{2}}{4}$

Step 3.3.2.1.7

Factor $x$ out of $x^{4} + 2 x^{3} + x^{2} + 2 K x + 2 K x^{2}$ .

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Step 3.3.2.1.7.1

Factor $x$ out of $x^{4}$ .

$y = \frac{x \cdot x^{3} + 2 x^{3} + x^{2} + 2 K x + 2 K x^{2}}{4} + \frac{K^{2}}{4}$

Step 3.3.2.1.7.2

Factor $x$ out of $2 x^{3}$ .

$y = \frac{x \cdot x^{3} + x (2 x^{2}) + x^{2} + 2 K x + 2 K x^{2}}{4} + \frac{K^{2}}{4}$

Step 3.3.2.1.7.3

Factor $x$ out of $x^{2}$ .

$y = \frac{x \cdot x^{3} + x (2 x^{2}) + x \cdot x + 2 K x + 2 K x^{2}}{4} + \frac{K^{2}}{4}$

Step 3.3.2.1.7.4

Factor $x$ out of $2 K x$ .

$y = \frac{x \cdot x^{3} + x (2 x^{2}) + x \cdot x + x (2 K) + 2 K x^{2}}{4} + \frac{K^{2}}{4}$

Step 3.3.2.1.7.5

Factor $x$ out of $2 K x^{2}$ .

$y = \frac{x \cdot x^{3} + x (2 x^{2}) + x \cdot x + x (2 K) + x (2 K x)}{4} + \frac{K^{2}}{4}$

Step 3.3.2.1.7.6

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

$y = \frac{x (x^{3} + 2 x^{2}) + x \cdot x + x (2 K) + x (2 K x)}{4} + \frac{K^{2}}{4}$

Step 3.3.2.1.7.7

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

$y = \frac{x (x^{3} + 2 x^{2} + x) + x (2 K) + x (2 K x)}{4} + \frac{K^{2}}{4}$

Step 3.3.2.1.7.8

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

$y = \frac{x (x^{3} + 2 x^{2} + x + 2 K) + x (2 K x)}{4} + \frac{K^{2}}{4}$

Step 3.3.2.1.7.9

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

$y = \frac{x (x^{3} + 2 x^{2} + x + 2 K + 2 K x)}{4} + \frac{K^{2}}{4}$

Step 3.4

Simplify $\frac{x (x^{3} + 2 x^{2} + x + 2 K + 2 K x)}{4} + \frac{K^{2}}{4}$ .

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Step 3.4.1

Move $x$ .

$y = \frac{x (x^{3} + 2 x^{2} + 2 K + 2 K x + x)}{4} + \frac{K^{2}}{4}$

Step 3.4.2

Move $2 K$ .

$y = \frac{x (x^{3} + 2 x^{2} + 2 K x + 2 K + x)}{4} + \frac{K^{2}}{4}$

Step 3.4.3

Move $2 x^{2}$ .

$y = \frac{x (x^{3} + 2 K x + 2 x^{2} + 2 K + x)}{4} + \frac{K^{2}}{4}$

Step 4

Simplify the constant of integration.

$y = \frac{x (x^{3} + K x + 2 x^{2} + K + x)}{4} + K$

Step 5

Use the initial condition to find the value of $K$ by substituting $0$ for $x$ and $1$ for $y$ in $y = \frac{x (x^{3} + K x + 2 x^{2} + K + x)}{4} + K$ .

$1 = \frac{0 (0^{3} + K \cdot 0 + 2 \cdot 0^{2} + K + 0)}{4} + K$

Step 6

Solve for $K$ .

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Step 6.1

Rewrite the equation as $\frac{0 (0^{3} + K \cdot 0 + 2 \cdot 0^{2} + K + 0)}{4} + K = 1$ .

$\frac{0 (0^{3} + K \cdot 0 + 2 \cdot 0^{2} + K + 0)}{4} + K = 1$

Step 6.2

Simplify $\frac{0 (0^{3} + K \cdot 0 + 2 \cdot 0^{2} + K + 0)}{4} + K$ .

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Step 6.2.1

Add $0^{3} + K \cdot 0 + 2 \cdot 0^{2} + K$ and $0$ .

$\frac{0 (0^{3} + K \cdot 0 + 2 \cdot 0^{2} + K)}{4} + K = 1$

Step 6.2.2

Add $K \cdot 0$ and $K$ .

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Step 6.2.2.1

Reorder $0 \cdot K$ and $K$ .

$\frac{0 (0^{3} + 2 \cdot 0^{2} + K + 0 \cdot K)}{4} + K = 1$

Step 6.2.2.2

Add $K$ and $0 \cdot K$ .

$\frac{0 (0^{3} + 2 \cdot 0^{2} + K)}{4} + K = 1$

Step 6.2.3

Simplify each term.

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Step 6.2.3.1

Cancel the common factor of $0$ and $4$ .

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Step 6.2.3.1.1

Factor $4$ out of $0 (0^{3} + 2 \cdot 0^{2} + K)$ .

$\frac{4 (0 (0^{3} + 2 \cdot 0^{2} + K))}{4} + K = 1$

Step 6.2.3.1.2

Cancel the common factors.

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Step 6.2.3.1.2.1

Factor $4$ out of $4$ .

$\frac{4 (0 (0^{3} + 2 \cdot 0^{2} + K))}{4 (1)} + K = 1$

Step 6.2.3.1.2.2

Cancel the common factor.

$\frac{4 (0 (0^{3} + 2 \cdot 0^{2} + K))}{4 \cdot 1} + K = 1$

Step 6.2.3.1.2.3

Rewrite the expression.

$\frac{0 (0^{3} + 2 \cdot 0^{2} + K)}{1} + K = 1$

Step 6.2.3.1.2.4

Divide $0 (0^{3} + 2 \cdot 0^{2} + K)$ by $1$ .

$0 (0^{3} + 2 \cdot 0^{2} + K) + K = 1$

Step 6.2.3.2

Simplify each term.

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Step 6.2.3.2.1

Raising $0$ to any positive power yields $0$ .

$0 (0 + 2 \cdot 0^{2} + K) + K = 1$

Step 6.2.3.2.2

Raising $0$ to any positive power yields $0$ .

$0 (0 + 2 \cdot 0 + K) + K = 1$

Step 6.2.3.2.3

Multiply $2$ by $0$ .

$0 (0 + 0 + K) + K = 1$

Step 6.2.3.3

Combine the opposite terms in $0 + 0 + K$ .

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Step 6.2.3.3.1

Add $0$ and $0$ .

$0 (0 + K) + K = 1$

Step 6.2.3.3.2

Add $0$ and $K$ .

$0 K + K = 1$

Step 6.2.3.4

Multiply $0$ by $K$ .

$0 + K = 1$

Step 6.2.4

Add $0$ and $K$ .

$K = 1$

Step 7

Substitute $1$ for $K$ in $y = \frac{x (x^{3} + K x + 2 x^{2} + K + x)}{4} + K$ and simplify.

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Step 7.1

Substitute $1$ for $K$ .

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

Step 7.2

Add $1 x$ and $x$ .

$y = \frac{x (x^{3} + 2 x^{2} + K + 2 x)}{4} + K$

Step 7.3

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

$y = \frac{x (x^{3} + 2 x^{2} + K + 2 x)}{4} + K \cdot \frac{4}{4}$

Step 7.4

Combine $K$ and $\frac{4}{4}$ .

$y = \frac{x (x^{3} + 2 x^{2} + K + 2 x)}{4} + \frac{K \cdot 4}{4}$

Step 7.5

Combine the numerators over the common denominator.

$y = \frac{x (x^{3} + 2 x^{2} + K + 2 x) + K \cdot 4}{4}$

Step 7.6

Simplify the numerator.

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Step 7.6.1

Apply the distributive property.

$y = \frac{x \cdot x^{3} + x (2 x^{2}) + x K + x (2 x) + K \cdot 4}{4}$

Step 7.6.2

Simplify.

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Step 7.6.2.1

Multiply $x$ by $x^{3}$ by adding the exponents.

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Step 7.6.2.1.1

Multiply $x$ by $x^{3}$ .

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Step 7.6.2.1.1.1

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

$y = \frac{x^{1} x^{3} + x (2 x^{2}) + x K + x (2 x) + K \cdot 4}{4}$

Step 7.6.2.1.1.2

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

$y = \frac{x^{1 + 3} + x (2 x^{2}) + x K + x (2 x) + K \cdot 4}{4}$

Step 7.6.2.1.2

Add $1$ and $3$ .

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

Step 7.6.2.2

Rewrite using the commutative property of multiplication.

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

Step 7.6.2.3

Rewrite using the commutative property of multiplication.

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

Step 7.6.3

Simplify each term.

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Step 7.6.3.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 7.6.3.1.1

Move $x^{2}$ .

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

Step 7.6.3.1.2

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

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Step 7.6.3.1.2.1

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

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

Step 7.6.3.1.2.2

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

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

Step 7.6.3.1.3

Add $2$ and $1$ .

$y = \frac{x^{4} + 2 x^{3} + x K + 2 x \cdot x + K \cdot 4}{4}$

Step 7.6.3.2

Multiply $x$ by $x$ by adding the exponents.

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Step 7.6.3.2.1

Move $x$ .

$y = \frac{x^{4} + 2 x^{3} + x K + 2 (x \cdot x) + K \cdot 4}{4}$

Step 7.6.3.2.2

Multiply $x$ by $x$ .

$y = \frac{x^{4} + 2 x^{3} + x K + 2 x^{2} + K \cdot 4}{4}$

Step 7.6.4

Move $4$ to the left of $K$ .

$y = \frac{x^{4} + 2 x^{3} + x K + 2 x^{2} + 4 K}{4}$