Calculus Examples

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

Step 1

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

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

Step 2

Simplify the answer.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Reorder terms.

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

Step 3

Simplify.

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Step 3.1

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

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

Step 3.2

Divide $1$ by $1$ .

$\ln (1 | x |)$

Step 3.3

Multiply $| x |$ by $1$ .

$\ln (| x |)$