Calculus Examples

$\int \sqrt{\frac{x^{2}}{2}} d x$

Step 1

Let $u_{2} = \sqrt{\frac{x^{2}}{2}}$ . Then $d u_{2} = \frac{1}{2^{\frac{1}{2}}} d x$ , so $2^{\frac{1}{2}} d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 1.1

Let $u_{2} = \sqrt{\frac{x^{2}}{2}}$ . Find $\frac{d u_{2}}{d x}$ .

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Step 1.1.1

Differentiate $\sqrt{\frac{x^{2}}{2}}$ .

$\frac{d}{d x} [\sqrt{\frac{x^{2}}{2}}]$

Step 1.1.2

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

$\frac{d}{d x} [{(\frac{x^{2}}{2})}^{\frac{1}{2}}]$

Step 1.1.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = \frac{x^{2}}{2}$ .

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Step 1.1.3.1

To apply the Chain Rule, set $u_{1}$ as $\frac{x^{2}}{2}$ .

$\frac{d}{d u_{1}} [{u_{1}}^{\frac{1}{2}}] \frac{d}{d x} [\frac{x^{2}}{2}]$

Step 1.1.3.2

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

$\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d x} [\frac{x^{2}}{2}]$

Step 1.1.3.3

Replace all occurrences of $u_{1}$ with $\frac{x^{2}}{2}$ .

$\frac{1}{2} {(\frac{x^{2}}{2})}^{\frac{1}{2} - 1} \frac{d}{d x} [\frac{x^{2}}{2}]$

Step 1.1.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} {(\frac{x^{2}}{2})}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [\frac{x^{2}}{2}]$

Step 1.1.5

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

$\frac{1}{2} {(\frac{x^{2}}{2})}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [\frac{x^{2}}{2}]$

Step 1.1.6

Combine the numerators over the common denominator.

$\frac{1}{2} {(\frac{x^{2}}{2})}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [\frac{x^{2}}{2}]$

Step 1.1.7

Simplify the numerator.

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Step 1.1.7.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} {(\frac{x^{2}}{2})}^{\frac{1 - 2}{2}} \frac{d}{d x} [\frac{x^{2}}{2}]$

Step 1.1.7.2

Subtract $2$ from $1$ .

$\frac{1}{2} {(\frac{x^{2}}{2})}^{\frac{- 1}{2}} \frac{d}{d x} [\frac{x^{2}}{2}]$

Step 1.1.8

Move the negative in front of the fraction.

$\frac{1}{2} {(\frac{x^{2}}{2})}^{- \frac{1}{2}} \frac{d}{d x} [\frac{x^{2}}{2}]$

Step 1.1.9

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x^{2}}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x^{2}]$ .

$\frac{1}{2} {(\frac{x^{2}}{2})}^{- \frac{1}{2}} (\frac{1}{2} \frac{d}{d x} [x^{2}])$

Step 1.1.10

Combine fractions.

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Step 1.1.10.1

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

$\frac{1}{2 \cdot 2} {(\frac{x^{2}}{2})}^{- \frac{1}{2}} \frac{d}{d x} [x^{2}]$

Step 1.1.10.2

Multiply $2$ by $2$ .

$\frac{1}{4} {(\frac{x^{2}}{2})}^{- \frac{1}{2}} \frac{d}{d x} [x^{2}]$

Step 1.1.11

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

$\frac{1}{4} {(\frac{x^{2}}{2})}^{- \frac{1}{2}} (2 x)$

Step 1.1.12

Simplify terms.

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Step 1.1.12.1

Combine $2$ and $\frac{1}{4}$ .

$\frac{2}{4} {(\frac{x^{2}}{2})}^{- \frac{1}{2}} x$

Step 1.1.12.2

Combine $x$ and $\frac{2}{4}$ .

$\frac{x \cdot 2}{4} {(\frac{x^{2}}{2})}^{- \frac{1}{2}}$

Step 1.1.12.3

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

$\frac{2 \cdot x}{4} {(\frac{x^{2}}{2})}^{- \frac{1}{2}}$

Step 1.1.12.4

Cancel the common factor of $2$ and $4$ .

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Step 1.1.12.4.1

Factor $2$ out of $2 x$ .

$\frac{2 (x)}{4} {(\frac{x^{2}}{2})}^{- \frac{1}{2}}$

Step 1.1.12.4.2

Cancel the common factors.

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Step 1.1.12.4.2.1

Factor $2$ out of $4$ .

$\frac{2 x}{2 \cdot 2} {(\frac{x^{2}}{2})}^{- \frac{1}{2}}$

Step 1.1.12.4.2.2

Cancel the common factor.

$\frac{2 x}{2 \cdot 2} {(\frac{x^{2}}{2})}^{- \frac{1}{2}}$

Step 1.1.12.4.2.3

Rewrite the expression.

$\frac{x}{2} {(\frac{x^{2}}{2})}^{- \frac{1}{2}}$

Step 1.1.13

Simplify.

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Step 1.1.13.1

Change the sign of the exponent by rewriting the base as its reciprocal.

$\frac{x}{2} {(\frac{2}{x^{2}})}^{\frac{1}{2}}$

Step 1.1.13.2

Apply the product rule to $\frac{2}{x^{2}}$ .

$\frac{x}{2} \cdot \frac{2^{\frac{1}{2}}}{{(x^{2})}^{\frac{1}{2}}}$

Step 1.1.13.3

Combine terms.

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Step 1.1.13.3.1

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

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Step 1.1.13.3.1.1

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

$\frac{x}{2} \cdot \frac{2^{\frac{1}{2}}}{x^{2 (\frac{1}{2})}}$

Step 1.1.13.3.1.2

Cancel the common factor of $2$ .

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Step 1.1.13.3.1.2.1

Cancel the common factor.

$\frac{x}{2} \cdot \frac{2^{\frac{1}{2}}}{x^{2 (\frac{1}{2})}}$

Step 1.1.13.3.1.2.2

Rewrite the expression.

$\frac{x}{2} \cdot \frac{2^{\frac{1}{2}}}{x^{1}}$

Step 1.1.13.3.2

Simplify.

$\frac{x}{2} \cdot \frac{2^{\frac{1}{2}}}{x}$

Step 1.1.13.3.3

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

$\frac{x \cdot 2^{\frac{1}{2}}}{2 x}$

Step 1.1.13.3.4

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

$\frac{x}{2 x \cdot 2^{- \frac{1}{2}}}$

Step 1.1.13.3.5

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

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Step 1.1.13.3.5.1

Move $2^{- \frac{1}{2}}$ .

$\frac{x}{2^{- \frac{1}{2}} \cdot 2 x}$

Step 1.1.13.3.5.2

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

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Step 1.1.13.3.5.2.1

Raise $2$ to the power of $1$ .

$\frac{x}{2^{- \frac{1}{2}} \cdot 2^{1} x}$

Step 1.1.13.3.5.2.2

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

$\frac{x}{2^{- \frac{1}{2} + 1} x}$

Step 1.1.13.3.5.3

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

$\frac{x}{2^{- \frac{1}{2} + \frac{2}{2}} x}$

Step 1.1.13.3.5.4

Combine the numerators over the common denominator.

$\frac{x}{2^{\frac{- 1 + 2}{2}} x}$

Step 1.1.13.3.5.5

Add $- 1$ and $2$ .

$\frac{x}{2^{\frac{1}{2}} x}$

Step 1.1.13.3.6

Cancel the common factor.

$\frac{x}{2^{\frac{1}{2}} x}$

Step 1.1.13.3.7

Rewrite the expression.

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

Step 1.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\int u_{2} \frac{1}{\frac{1}{2^{\frac{1}{2}}}} d u_{2}$

Step 2

Simplify.

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Step 2.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{2^{\frac{1}{2}}}$ .

$\int u_{2} (1 \cdot 2^{\frac{1}{2}}) d u_{2}$

Step 2.2

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

$\int u_{2} \cdot 2^{\frac{1}{2}} d u_{2}$

Step 3

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

$2^{\frac{1}{2}} \int u_{2} d u_{2}$

Step 4

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

$2^{\frac{1}{2}} (\frac{1}{2} {u_{2}}^{2} + C)$

Step 5

Simplify.

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Step 5.1

Rewrite $2^{\frac{1}{2}} (\frac{1}{2} {u_{2}}^{2} + C)$ as $2^{\frac{1}{2}} \frac{1}{2} {u_{2}}^{2} + C$ .

$2^{\frac{1}{2}} \frac{1}{2} {u_{2}}^{2} + C$

Step 5.2

Simplify.

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Step 5.2.1

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

$\frac{2^{\frac{1}{2}}}{2} {u_{2}}^{2} + C$

Step 5.2.2

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

$\frac{1}{2 \cdot 2^{- \frac{1}{2}}} {u_{2}}^{2} + C$

Step 5.2.3

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

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Step 5.2.3.1

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

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Step 5.2.3.1.1

Raise $2$ to the power of $1$ .

$\frac{1}{2^{1} \cdot 2^{- \frac{1}{2}}} {u_{2}}^{2} + C$

Step 5.2.3.1.2

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

$\frac{1}{2^{1 - \frac{1}{2}}} {u_{2}}^{2} + C$

Step 5.2.3.2

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

$\frac{1}{2^{\frac{2}{2} - \frac{1}{2}}} {u_{2}}^{2} + C$

Step 5.2.3.3

Combine the numerators over the common denominator.

$\frac{1}{2^{\frac{2 - 1}{2}}} {u_{2}}^{2} + C$

Step 5.2.3.4

Subtract $1$ from $2$ .

$\frac{1}{2^{\frac{1}{2}}} {u_{2}}^{2} + C$

Step 6

Replace all occurrences of $u_{2}$ with $\sqrt{\frac{x^{2}}{2}}$ .

$\frac{1}{2^{\frac{1}{2}}} {\sqrt{\frac{x^{2}}{2}}}^{2} + C$

Step 7

Reorder terms.

$\frac{1}{2^{\frac{1}{2}}} {\sqrt{\frac{1}{2} x^{2}}}^{2} + C$