Calculus Examples

$\int \ln ({(x)}^{8}) \cdot \frac{1}{x} d x$

Step 1

Combine $\ln ({(x)}^{8})$ and $\frac{1}{x}$ .

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

Step 2

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

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

Step 3

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

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

Step 4

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 4.1

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

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Step 4.1.1

Differentiate $\ln (x)$ .

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

Step 4.1.2

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

$\frac{1}{x}$

Step 4.2

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

$8 \int u d u$

Step 5

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

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

Simplify.

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Step 6.2.1

Combine $8$ and $\frac{1}{2}$ .

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

Step 6.2.2

Cancel the common factor of $8$ and $2$ .

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Step 6.2.2.1

Factor $2$ out of $8$ .

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

Step 6.2.2.2

Cancel the common factors.

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Step 6.2.2.2.1

Factor $2$ out of $2$ .

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

Step 6.2.2.2.2

Cancel the common factor.

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

Step 6.2.2.2.3

Rewrite the expression.

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

Step 6.2.2.2.4

Divide $4$ by $1$ .

$4 u^{2} + C$

Step 7

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

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