Calculus Examples

$\int \frac{\ln (x^{3})}{8 x} d x$

Step 1

Rewrite $\frac{\ln (x^{3})}{8 x}$ as $\frac{3 \ln (x)}{8 x}$ .

$\int \frac{3 \ln (x)}{8 x} d x$

Step 2

Since $\frac{3}{8}$ is constant with respect to $x$ , move $\frac{3}{8}$ out of the integral.

$\frac{3}{8} \int \frac{\ln (x)}{x} d x$

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 $d u$ .

$\frac{3}{8} \int u d u$

Step 4

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

$\frac{3}{8} (\frac{1}{2} u^{2} + C)$

Step 5

Simplify.

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Step 5.1

Rewrite $\frac{3}{8} (\frac{1}{2} u^{2} + C)$ as $\frac{3}{8} \cdot \frac{1}{2} u^{2} + C$ .

$\frac{3}{8} \cdot \frac{1}{2} u^{2} + C$

Step 5.2

Simplify.

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Step 5.2.1

Multiply $\frac{3}{8}$ by $\frac{1}{2}$ .

$\frac{3}{8 \cdot 2} u^{2} + C$

Step 5.2.2

Multiply $8$ by $2$ .

$\frac{3}{16} u^{2} + C$

Step 6

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

$\frac{3}{16} \ln^{2} (x) + C$