Calculus Examples

$\int_{\frac{π}{6}}^{\frac{π}{2}} \cot (x) d x$

Step 1

The integral of $\cot (x)$ with respect to $x$ is $\ln (| \sin (x) |)$ .

$\ln (| \sin (x) |)]_{\frac{π}{6}}^{\frac{π}{2}}$

Step 2

Simplify the answer.

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Step 2.1

Evaluate $\ln (| \sin (x) |)$ at $\frac{π}{2}$ and at $\frac{π}{6}$ .

$\ln (| \sin (\frac{π}{2}) |) - \ln (| \sin (\frac{π}{6}) |)$

Step 2.2

Simplify.

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Step 2.2.1

The exact value of $\sin (\frac{π}{2})$ is $1$ .

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

Step 2.2.2

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

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

Step 2.2.3

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

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

Step 2.3

Simplify.

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Step 2.3.1

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

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

Step 2.3.2

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\ln (\frac{1}{\frac{1}{2}})$

Step 2.3.3

Multiply the numerator by the reciprocal of the denominator.

$\ln (1 \cdot 2)$

Step 2.3.4

Multiply $2$ by $1$ .

$\ln (2)$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\ln (2)$

Decimal Form:

$0.69314718 \dots$