Calculus Examples

$\int_{1}^{6} - \frac{6}{x} d x$

Step 1

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$- \int_{1}^{6} \frac{6}{x} d x$

Step 2

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$- (6 \int_{1}^{6} \frac{1}{x} d x)$

Step 3

Multiply $6$ by $- 1$ .

$- 6 \int_{1}^{6} \frac{1}{x} d x$

Step 4

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

$- 6 (\ln (| x |)]_{1}^{6})$

Step 5

Simplify the answer.

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Step 5.1

Evaluate $\ln (| x |)$ at $6$ and at $1$ .

$- 6 (\ln (| 6 |) - \ln (| 1 |))$

Step 5.2

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

$- 6 \ln (\frac{| 6 |}{| 1 |})$

Step 5.3

Simplify.

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Step 5.3.1

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

$- 6 \ln (\frac{6}{| 1 |})$

Step 5.3.2

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

$- 6 \ln (\frac{6}{1})$

Step 5.3.3

Divide $6$ by $1$ .

$- 6 \ln (6)$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- 6 \ln (6)$

Decimal Form:

$- 10.75055681 \dots$

Step 7