Calculus Examples

$\int_{1}^{4} \frac{4}{x} d x$

Step 1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 \int_{1}^{4} \frac{1}{x} d x$

Step 2

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

$4 (\ln (| x |)]_{1}^{4})$

Step 3

Simplify the answer.

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Step 3.1

Evaluate $\ln (| x |)$ at $4$ and at $1$ .

$4 (\ln (| 4 |) - \ln (| 1 |))$

Step 3.2

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

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

Step 3.3

Simplify.

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Step 3.3.1

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

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

Step 3.3.2

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$4 \ln (\frac{4}{1})$

Step 3.3.3

Divide $4$ by $1$ .

$4 \ln (4)$

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

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

$4 (2 \ln (2))$

Step 4.3

Multiply $2$ by $4$ .

$8 \ln (2)$

Step 5

The result can be shown in multiple forms.

Exact Form:

$8 \ln (2)$

Decimal Form:

$5.54517744 \dots$

Step 6