Calculus Examples

$\int_{4}^{8} \frac{1}{x \ln (x^{2})} d x$

Step 1

Rewrite $\frac{1}{x \ln (x^{2})}$ as $\frac{1}{x (2 \ln (x))}$ .

$\int_{4}^{8} \frac{1}{x (2 \ln (x))} d x$

Step 2

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

$\frac{1}{2} \int_{4}^{8} \frac{1}{x (\ln (x))} d x$

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

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

$u_{lower} = \ln (4)$

Step 3.3

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

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

Step 3.4

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

$u_{lower} = \ln (4)$

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

Step 3.5

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

$\frac{1}{2} \int_{\ln (4)}^{\ln (8)} \frac{1}{u} d u$

Step 4

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$\frac{1}{2} \ln (| u |)]_{\ln (4)}^{\ln (8)}$

Step 5

Evaluate $\ln (| u |)$ at $\ln (8)$ and at $\ln (4)$ .

$\frac{1}{2} (\ln (| \ln (8) |) - \ln (| \ln (4) |))$

Step 6

Simplify.

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Step 6.1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\frac{1}{2} \ln (\frac{| \ln (8) |}{| \ln (4) |})$

Step 6.2

Combine $\frac{1}{2}$ and $\ln (\frac{| \ln (8) |}{| \ln (4) |})$ .

$\frac{\ln (\frac{| \ln (8) |}{| \ln (4) |})}{2}$

Step 7

Simplify.

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Step 7.1

$\ln (8)$ is approximately $2.07944154$ which is positive so remove the absolute value

$\frac{\ln (\frac{\ln (8)}{| \ln (4) |})}{2}$

Step 7.2

$\ln (4)$ is approximately $1.38629436$ which is positive so remove the absolute value

$\frac{\ln (\frac{\ln (8)}{\ln (4)})}{2}$

Step 7.3

Rewrite $\ln (8)$ as $\ln (2^{3})$ .

$\frac{\ln (\frac{\ln (2^{3})}{\ln (4)})}{2}$

Step 7.4

Expand $\ln (2^{3})$ by moving $3$ outside the logarithm.

$\frac{\ln (\frac{3 \ln (2)}{\ln (4)})}{2}$

Step 7.5

Rewrite $\ln (4)$ as $\ln (2^{2})$ .

$\frac{\ln (\frac{3 \ln (2)}{\ln (2^{2})})}{2}$

Step 7.6

Expand $\ln (2^{2})$ by moving $2$ outside the logarithm.

$\frac{\ln (\frac{3 \ln (2)}{2 \ln (2)})}{2}$

Step 7.7

Cancel the common factor of $\ln (2)$ .

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Step 7.7.1

Cancel the common factor.

$\frac{\ln (\frac{3 \ln (2)}{2 \ln (2)})}{2}$

Step 7.7.2

Rewrite the expression.

$\frac{\ln (\frac{3}{2})}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{\ln (\frac{3}{2})}{2}$

Decimal Form:

$0.20273255 \dots$