Calculus Examples

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

Step 1

Separate the variables.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Cancel the common factor of $2$ .

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Step 1.1.2.1.1

Cancel the common factor.

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

Step 1.1.2.1.2

Rewrite the expression.

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

Step 1.1.2.2

Cancel the common factor of $y$ .

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Step 1.1.2.2.1

Cancel the common factor.

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

Step 1.1.2.2.2

Rewrite the expression.

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

Step 1.1.2.3

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

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Step 1.1.2.3.1

Cancel the common factor.

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

Step 1.1.2.3.2

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

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

Step 1.1.3

Simplify the right side.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

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

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Step 1.1.3.1.1.1

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

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

Step 1.1.3.1.1.2

Cancel the common factors.

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Step 1.1.3.1.1.2.1

Factor $y$ out of $2 y \sqrt{x}$ .

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

Step 1.1.3.1.1.2.2

Cancel the common factor.

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

Step 1.1.3.1.1.2.3

Rewrite the expression.

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

Step 1.1.3.1.2

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

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

Step 1.1.3.1.3

Combine and simplify the denominator.

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Step 1.1.3.1.3.1

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

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

Step 1.1.3.1.3.2

Move $\sqrt{x}$ .

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

Step 1.1.3.1.3.3

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

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

Step 1.1.3.1.3.4

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

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

Step 1.1.3.1.3.5

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

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

Step 1.1.3.1.3.6

Add $1$ and $1$ .

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

Step 1.1.3.1.3.7

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

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Step 1.1.3.1.3.7.1

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

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

Step 1.1.3.1.3.7.2

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

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

Step 1.1.3.1.3.7.3

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

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

Step 1.1.3.1.3.7.4

Cancel the common factor of $2$ .

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Step 1.1.3.1.3.7.4.1

Cancel the common factor.

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

Step 1.1.3.1.3.7.4.2

Rewrite the expression.

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

Step 1.1.3.1.3.7.5

Simplify.

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

Step 1.1.3.1.4

Cancel the common factor of $2$ .

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Step 1.1.3.1.4.1

Cancel the common factor.

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

Step 1.1.3.1.4.2

Rewrite the expression.

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

Step 1.1.3.1.5

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

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

Step 1.1.3.1.6

Combine and simplify the denominator.

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Step 1.1.3.1.6.1

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

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

Step 1.1.3.1.6.2

Move $\sqrt{x}$ .

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

Step 1.1.3.1.6.3

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

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

Step 1.1.3.1.6.4

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

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

Step 1.1.3.1.6.5

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

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

Step 1.1.3.1.6.6

Add $1$ and $1$ .

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

Step 1.1.3.1.6.7

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

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Step 1.1.3.1.6.7.1

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

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

Step 1.1.3.1.6.7.2

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

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

Step 1.1.3.1.6.7.3

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

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

Step 1.1.3.1.6.7.4

Cancel the common factor of $2$ .

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Step 1.1.3.1.6.7.4.1

Cancel the common factor.

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

Step 1.1.3.1.6.7.4.2

Rewrite the expression.

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

Step 1.1.3.1.6.7.5

Simplify.

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

Step 1.2

Factor.

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Step 1.2.1

To write $\frac{y \sqrt{x}}{2 x}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

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

Step 1.2.2

To write $\frac{\sqrt{x}}{y x}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.2.3

Write each expression with a common denominator of $2 x y$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.2.3.1

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

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

Step 1.2.3.2

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

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

Step 1.2.3.3

Reorder the factors of $y x \cdot 2$ .

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

Step 1.2.4

Combine the numerators over the common denominator.

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

Step 1.2.5

Simplify the numerator.

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Step 1.2.5.1

Factor $\sqrt{x}$ out of $y \sqrt{x} y + \sqrt{x} \cdot 2$ .

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Step 1.2.5.1.1

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

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

Step 1.2.5.1.2

Factor $\sqrt{x}$ out of $\sqrt{x} \cdot 2$ .

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

Step 1.2.5.1.3

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

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

Step 1.2.5.2

Multiply $y$ by $y$ .

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

Step 1.3

Regroup factors.

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

Step 1.4

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

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

Step 1.5

Simplify.

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Step 1.5.1

Multiply $\frac{\sqrt{x}}{2 x}$ by $\frac{y^{2} + 2}{y}$ .

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

Step 1.5.2

Cancel the common factor of $y$ .

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Step 1.5.2.1

Factor $y$ out of $2 x y$ .

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

Step 1.5.2.2

Cancel the common factor.

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

Step 1.5.2.3

Rewrite the expression.

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

Step 1.5.3

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

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Step 1.5.3.1

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

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

Step 1.5.3.2

Cancel the common factor.

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

Step 1.5.3.3

Rewrite the expression.

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

Step 1.6

Rewrite the equation.

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

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

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

Step 2.2

Integrate the left side.

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Step 2.2.1

Let $u = y^{2} + 2$ . Then $d u = 2 y d y$ , so $\frac{1}{2} d u = y d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.1.1

Let $u = y^{2} + 2$ . Find $\frac{d u}{d y}$ .

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Step 2.2.1.1.1

Differentiate $y^{2} + 2$ .

$\frac{d}{d y} [y^{2} + 2]$

Step 2.2.1.1.2

By the Sum Rule, the derivative of $y^{2} + 2$ with respect to $y$ is $\frac{d}{d y} [y^{2}] + \frac{d}{d y} [2]$ .

$\frac{d}{d y} [y^{2}] + \frac{d}{d y} [2]$

Step 2.2.1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 2$ .

$2 y + \frac{d}{d y} [2]$

Step 2.2.1.1.4

Since $2$ is constant with respect to $y$ , the derivative of $2$ with respect to $y$ is $0$ .

$2 y + 0$

Step 2.2.1.1.5

Add $2 y$ and $0$ .

$2 y$

Step 2.2.1.2

Rewrite the problem using $u$ and $d u$ .

$\int \frac{1}{u} \cdot \frac{1}{2} d u = \int \frac{\sqrt{x}}{2 x} d x$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Multiply $\frac{1}{u}$ by $\frac{1}{2}$ .

$\int \frac{1}{u \cdot 2} d u = \int \frac{\sqrt{x}}{2 x} d x$

Step 2.2.2.2

Move $2$ to the left of $u$ .

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

Step 2.2.3

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

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

Step 2.2.4

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

$\frac{1}{2} (\ln (| u |) + C_{1}) = \int \frac{\sqrt{x}}{2 x} d x$

Step 2.2.5

Simplify.

$\frac{1}{2} \ln (| u |) + C_{1} = \int \frac{\sqrt{x}}{2 x} d x$

Step 2.2.6

Replace all occurrences of $u$ with $y^{2} + 2$ .

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

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

Step 2.3.2

Simplify the expression.

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Step 2.3.2.1

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

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

Step 2.3.2.2

Simplify.

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Step 2.3.2.2.1

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

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

Step 2.3.2.2.2

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

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Step 2.3.2.2.2.1

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

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Step 2.3.2.2.2.1.1

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

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

Step 2.3.2.2.2.1.2

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

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

Step 2.3.2.2.2.2

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

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

Step 2.3.2.2.2.3

Combine the numerators over the common denominator.

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

Step 2.3.2.2.2.4

Subtract $1$ from $2$ .

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

Step 2.3.2.3

Apply basic rules of exponents.

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Step 2.3.2.3.1

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

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

Step 2.3.2.3.2

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

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Step 2.3.2.3.2.1

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

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

Step 2.3.2.3.2.2

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

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

Step 2.3.2.3.2.3

Move the negative in front of the fraction.

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

Step 2.3.3

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

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

Step 2.3.4

Simplify the answer.

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Step 2.3.4.1

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

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

Step 2.3.4.2

Simplify.

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Step 2.3.4.2.1

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

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

Step 2.3.4.2.2

Cancel the common factor of $2$ .

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Step 2.3.4.2.2.1

Cancel the common factor.

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

Step 2.3.4.2.2.2

Rewrite the expression.

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

Step 2.3.4.2.3

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

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

Step 2.4

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

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