Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Simplify.

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Step 3.3.1

$e$ is approximately $2.71828182$ which is positive so remove the absolute value

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

Step 3.3.2

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

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

Step 3.3.3

Divide $e$ by $1$ .

$4 \ln (e)$

Step 3.3.4

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

$4 \cdot 1$

Step 3.3.5

Multiply $4$ by $1$ .

$4$

Step 4