Calculus Examples

$\sqrt{x} d y = \sqrt{y} d x$

Step 1

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

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

Combine and simplify the denominator.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Add $1$ and $1$ .

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

Step 2.2.6

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

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Step 2.2.6.1

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

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

Step 2.2.6.2

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

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

Step 2.2.6.3

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

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

Step 2.2.6.4

Cancel the common factor of $2$ .

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Step 2.2.6.4.1

Cancel the common factor.

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

Step 2.2.6.4.2

Rewrite the expression.

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

Step 2.2.6.5

Simplify.

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

Step 2.3

Multiply $\frac{\sqrt{x y}}{x y} \sqrt{x}$ .

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Step 2.3.1

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

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

Step 2.3.2

Combine using the product rule for radicals.

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

Step 2.3.3

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

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

Step 2.3.4

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

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

Step 2.3.5

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

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

Step 2.3.6

Add $1$ and $1$ .

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

Step 2.4

Pull terms out from under the radical.

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

Step 2.5

Cancel the common factor of $x$ .

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Step 2.5.1

Cancel the common factor.

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

Step 2.5.2

Rewrite the expression.

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

Step 2.6

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

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

Step 2.7

Combine and simplify the denominator.

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Step 2.7.1

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

$\frac{\sqrt{y}}{y} d y = \frac{\sqrt{x y}}{\sqrt{x y} \sqrt{x y}} \sqrt{y} d x$

Step 2.7.2

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

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

Step 2.7.3

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

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

Step 2.7.4

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

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

Step 2.7.5

Add $1$ and $1$ .

$\frac{\sqrt{y}}{y} d y = \frac{\sqrt{x y}}{{\sqrt{x y}}^{2}} \sqrt{y} d x$

Step 2.7.6

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

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Step 2.7.6.1

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

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

Step 2.7.6.2

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

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

Step 2.7.6.3

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

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

Step 2.7.6.4

Cancel the common factor of $2$ .

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Step 2.7.6.4.1

Cancel the common factor.

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

Step 2.7.6.4.2

Rewrite the expression.

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

Step 2.7.6.5

Simplify.

$\frac{\sqrt{y}}{y} d y = \frac{\sqrt{x y}}{x y} \sqrt{y} d x$

Step 2.8

Multiply $\frac{\sqrt{x y}}{x y} \sqrt{y}$ .

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Step 2.8.1

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

$\frac{\sqrt{y}}{y} d y = \frac{\sqrt{x y} \sqrt{y}}{x y} d x$

Step 2.8.2

Combine using the product rule for radicals.

$\frac{\sqrt{y}}{y} d y = \frac{\sqrt{x y \cdot y}}{x y} d x$

Step 2.8.3

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

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

Step 2.8.4

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

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

Step 2.8.5

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

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

Step 2.8.6

Add $1$ and $1$ .

$\frac{\sqrt{y}}{y} d y = \frac{\sqrt{x y^{2}}}{x y} d x$

Step 2.9

Simplify the numerator.

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Step 2.9.1

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

$\frac{\sqrt{y}}{y} d y = \frac{\sqrt{y^{2} x}}{x y} d x$

Step 2.9.2

Pull terms out from under the radical.

$\frac{\sqrt{y}}{y} d y = \frac{y \sqrt{x}}{x y} d x$

Step 2.10

Cancel the common factor of $y$ .

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Step 2.10.1

Cancel the common factor.

$\frac{\sqrt{y}}{y} d y = \frac{y \sqrt{x}}{x y} d x$

Step 2.10.2

Rewrite the expression.

$\frac{\sqrt{y}}{y} d y = \frac{\sqrt{x}}{x} d x$

Step 3

Integrate both sides.

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Step 3.1

Set up an integral on each side.

$\int \frac{\sqrt{y}}{y} d y = \int \frac{\sqrt{x}}{x} d x$

Step 3.2

Integrate the left side.

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Step 3.2.1

Simplify the expression.

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Step 3.2.1.1

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

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

Step 3.2.1.2

Split the fraction into multiple fractions.

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Step 3.2.1.2.1

Move $y^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 3.2.1.2.2

Multiply $y$ by $y^{- \frac{1}{2}}$ by adding the exponents.

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Step 3.2.1.2.2.1

Multiply $y$ by $y^{- \frac{1}{2}}$ .

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Step 3.2.1.2.2.1.1

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

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

Step 3.2.1.2.2.1.2

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

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

Step 3.2.1.2.2.2

Write $1$ as a fraction with a common denominator.

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

Step 3.2.1.2.2.3

Combine the numerators over the common denominator.

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

Step 3.2.1.2.2.4

Subtract $1$ from $2$ .

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

Step 3.2.1.3

Apply basic rules of exponents.

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Step 3.2.1.3.1

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

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

Step 3.2.1.3.2

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

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Step 3.2.1.3.2.1

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

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

Step 3.2.1.3.2.2

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

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

Step 3.2.1.3.2.3

Move the negative in front of the fraction.

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

Step 3.2.2

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

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

Step 3.3

Integrate the right side.

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Step 3.3.1

Simplify the expression.

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Step 3.3.1.1

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

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

Step 3.3.1.2

Split the fraction into multiple fractions.

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Step 3.3.1.2.1

Move $x^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

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

Step 3.3.1.2.2

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 3.3.1.2.2.1

Multiply $x$ by $x^{- \frac{1}{2}}$ .

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Step 3.3.1.2.2.1.1

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

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

Step 3.3.1.2.2.1.2

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

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

Step 3.3.1.2.2.2

Write $1$ as a fraction with a common denominator.

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

Step 3.3.1.2.2.3

Combine the numerators over the common denominator.

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

Step 3.3.1.2.2.4

Subtract $1$ from $2$ .

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

Step 3.3.1.3

Apply basic rules of exponents.

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Step 3.3.1.3.1

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

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

Step 3.3.1.3.2

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

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Step 3.3.1.3.2.1

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

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

Step 3.3.1.3.2.2

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

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

Step 3.3.1.3.2.3

Move the negative in front of the fraction.

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

Step 3.3.2

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

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

Step 3.4

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

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

Step 4

Solve for $y$ .

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor.

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

Step 4.1.2.2

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

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

Step 4.1.3

Simplify the right side.

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Step 4.1.3.1

Simplify each term.

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Step 4.1.3.1.1

Cancel the common factor.

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

Step 4.1.3.1.2

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

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

Step 4.2

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

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

Step 4.3

Simplify the exponent.

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Step 4.3.1

Simplify the left side.

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Step 4.3.1.1

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

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Step 4.3.1.1.1

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

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Step 4.3.1.1.1.1

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

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

Step 4.3.1.1.1.2

Cancel the common factor of $2$ .

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Step 4.3.1.1.1.2.1

Cancel the common factor.

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

Step 4.3.1.1.1.2.2

Rewrite the expression.

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

Step 4.3.1.1.2

Simplify.

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

Step 4.3.2

Simplify the right side.

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Step 4.3.2.1

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

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Step 4.3.2.1.1

Rewrite ${(x^{\frac{1}{2}} + \frac{K}{2})}^{2}$ as $(x^{\frac{1}{2}} + \frac{K}{2}) (x^{\frac{1}{2}} + \frac{K}{2})$ .

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

Step 4.3.2.1.2

Expand $(x^{\frac{1}{2}} + \frac{K}{2}) (x^{\frac{1}{2}} + \frac{K}{2})$ using the FOIL Method.

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Step 4.3.2.1.2.1

Apply the distributive property.

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

Step 4.3.2.1.2.2

Apply the distributive property.

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

Step 4.3.2.1.2.3

Apply the distributive property.

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

Step 4.3.2.1.3

Simplify and combine like terms.

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Step 4.3.2.1.3.1

Simplify each term.

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Step 4.3.2.1.3.1.1

Multiply $x^{\frac{1}{2}}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 4.3.2.1.3.1.1.1

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

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

Step 4.3.2.1.3.1.1.2

Combine the numerators over the common denominator.

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

Step 4.3.2.1.3.1.1.3

Add $1$ and $1$ .

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

Step 4.3.2.1.3.1.1.4

Divide $2$ by $2$ .

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

Step 4.3.2.1.3.1.2

Simplify $x^{1}$ .

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

Step 4.3.2.1.3.1.3

Combine $x^{\frac{1}{2}}$ and $\frac{K}{2}$ .

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

Step 4.3.2.1.3.1.4

Combine $\frac{K}{2}$ and $x^{\frac{1}{2}}$ .

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

Step 4.3.2.1.3.1.5

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

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Step 4.3.2.1.3.1.5.1

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

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

Step 4.3.2.1.3.1.5.2

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

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

Step 4.3.2.1.3.1.5.3

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

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

Step 4.3.2.1.3.1.5.4

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

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

Step 4.3.2.1.3.1.5.5

Add $1$ and $1$ .

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

Step 4.3.2.1.3.1.5.6

Multiply $2$ by $2$ .

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

Step 4.3.2.1.3.2

Add $\frac{x^{\frac{1}{2}} K}{2}$ and $\frac{K x^{\frac{1}{2}}}{2}$ .

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Step 4.3.2.1.3.2.1

Reorder $x^{\frac{1}{2}}$ and $K$ .

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

Step 4.3.2.1.3.2.2

Add $\frac{K x^{\frac{1}{2}}}{2}$ and $\frac{K x^{\frac{1}{2}}}{2}$ .

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

Step 4.3.2.1.4

Cancel the common factor of $2$ .

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Step 4.3.2.1.4.1

Cancel the common factor.

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

Step 4.3.2.1.4.2

Rewrite the expression.

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

Step 4.4

Simplify $x + K x^{\frac{1}{2}} + \frac{K^{2}}{4}$ .

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Step 4.4.1

Move $K x^{\frac{1}{2}}$ .

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

Step 4.4.2

Reorder $x$ and $\frac{K^{2}}{4}$ .

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

Step 5

Simplify the constant of integration.

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