Calculus Examples

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

Step 1

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 1.1

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

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Step 1.1.1

Differentiate $\ln (x)$ .

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

Step 1.1.2

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

$\frac{1}{x}$

Step 1.2

Substitute the lower limit in for $x$ in $u = \ln (x)$ .

$u_{lower} = \ln (1)$

Step 1.3

The natural logarithm of $1$ is $0$ .

$u_{lower} = 0$

Step 1.4

Substitute the upper limit in for $x$ in $u = \ln (x)$ .

$u_{upper} = \ln (8)$

Step 1.5

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = \ln (8)$

Step 1.6

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{0}^{\ln (8)} u d u$

Step 2

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

$\frac{1}{2} u^{2}]_{0}^{\ln (8)}$

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $\frac{1}{2} u^{2}$ at $\ln (8)$ and at $0$ .

$(\frac{1}{2} \ln^{2} (8)) - \frac{1}{2} \cdot 0^{2}$

Step 3.2

Simplify.

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Step 3.2.1

Combine $\frac{1}{2}$ and $\ln^{2} (8)$ .

$\frac{\ln^{2} (8)}{2} - \frac{1}{2} \cdot 0^{2}$

Step 3.2.2

Raising $0$ to any positive power yields $0$ .

$\frac{\ln^{2} (8)}{2} - \frac{1}{2} \cdot 0$

Step 3.2.3

Multiply $0$ by $- 1$ .

$\frac{\ln^{2} (8)}{2} + 0 (\frac{1}{2})$

Step 3.2.4

Multiply $0$ by $\frac{1}{2}$ .

$\frac{\ln^{2} (8)}{2} + 0$

Step 3.2.5

Add $\frac{\ln^{2} (8)}{2}$ and $0$ .

$\frac{\ln^{2} (8)}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\ln^{2} (8)}{2}$

Decimal Form:

$2.16203856 \dots$