Calculus Examples

$\int \frac{\ln (x)}{2 x} d x$

Step 1

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

$\frac{1}{2} \int \frac{\ln (x)}{x} d x$

Step 2

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

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Step 2.1

Let $u = \ln (x)$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $\ln (x)$ .

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

Step 2.1.2

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

$\frac{1}{x}$

Step 2.2

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

$\frac{1}{2} \int u d u$

Step 3

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

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

Step 4

Simplify.

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Step 4.1

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

$\frac{1}{2} \cdot \frac{1}{2} u^{2} + C$

Step 4.2

Simplify.

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Step 4.2.1

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

$\frac{1}{2 \cdot 2} u^{2} + C$

Step 4.2.2

Multiply $2$ by $2$ .

$\frac{1}{4} u^{2} + C$

Step 5

Replace all occurrences of $u$ with $\ln (x)$ .

$\frac{1}{4} \ln^{2} (x) + C$