Calculus Examples

$\int \frac{\ln (\sqrt{x})}{x} d x$

Step 1

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

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Step 1.1

Let $u_{2} = \ln (\sqrt{x})$ . Find $\frac{d u_{2}}{d x}$ .

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Step 1.1.1

Differentiate $\ln (\sqrt{x})$ .

$\frac{d}{d x} [\ln (\sqrt{x})]$

Step 1.1.2

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

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

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 $f (x) = \ln (x)$ and $g (x) = x^{\frac{1}{2}}$ .

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Step 1.1.3.1

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

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

Step 1.1.3.2

The derivative of $\ln (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1}}$ .

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

Step 1.1.3.3

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

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

Step 1.1.4

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

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

Step 1.1.5

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

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

Step 1.1.6

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

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

Step 1.1.7

Combine the numerators over the common denominator.

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

Step 1.1.8

Simplify the numerator.

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Step 1.1.8.1

Multiply $- 1$ by $2$ .

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

Step 1.1.8.2

Subtract $2$ from $1$ .

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

Step 1.1.9

Move the negative in front of the fraction.

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

Step 1.1.10

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

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

Step 1.1.11

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

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

Step 1.1.12

Simplify the expression.

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Step 1.1.12.1

Move $2$ to the left of $x^{\frac{1}{2}}$ .

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

Step 1.1.12.2

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

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

Step 1.1.13

Simplify the denominator.

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Step 1.1.13.1

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

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Step 1.1.13.1.1

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

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

Step 1.1.13.1.2

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

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

Step 1.1.13.1.3

Combine the numerators over the common denominator.

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

Step 1.1.13.1.4

Add $1$ and $1$ .

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

Step 1.1.13.1.5

Divide $2$ by $2$ .

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

Step 1.1.13.2

Simplify $2 x^{1}$ .

$\frac{1}{2 x}$

Step 1.2

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

$\int 2 u_{2} d u_{2}$

Step 2

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

$2 \int u_{2} d u_{2}$

Step 3

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

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

${u_{2}}^{2} + C$

Step 5

Replace all occurrences of $u_{2}$ with $\ln (\sqrt{x})$ .

$\ln^{2} (\sqrt{x}) + C$