Calculus Examples

$\int \frac{\ln ({(x)}^{3})}{x} d x$

Step 1

Rewrite $\frac{\ln (x^{3})}{x}$ as $\frac{3 \ln (x)}{x}$ .

$\int \frac{3 \ln (x)}{x} d x$

Step 2

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int \frac{\ln (x)}{x} d x$

Step 3

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 3.1

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

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Step 3.1.1

Differentiate $\ln (x)$ .

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

Step 3.1.2

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

$\frac{1}{x}$

Step 3.2

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

$3 \int u d u$

Step 4

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

$3 (\frac{1}{2} u^{2} + C)$

Step 5

Simplify.

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Step 5.1

Rewrite $3 (\frac{1}{2} u^{2} + C)$ as $3 (\frac{1}{2}) u^{2} + C$ .

$3 (\frac{1}{2}) u^{2} + C$

Step 5.2

Combine $3$ and $\frac{1}{2}$ .

$\frac{3}{2} u^{2} + C$

Step 6

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

$\frac{3}{2} \ln^{2} (x) + C$