Calculus Examples

$\int_{\frac{π}{4}}^{\frac{π}{3}} \cot (x) d x$

Step 1

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

$\ln (| \sin (x) |)]_{\frac{π}{4}}^{\frac{π}{3}}$

Step 2

Simplify the answer.

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Step 2.1

Evaluate $\ln (| \sin (x) |)$ at $\frac{π}{3}$ and at $\frac{π}{4}$ .

$\ln (| \sin (\frac{π}{3}) |) - \ln (| \sin (\frac{π}{4}) |)$

Step 2.2

Simplify.

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Step 2.2.1

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$\ln (| \frac{\sqrt{3}}{2} |) - \ln (| \sin (\frac{π}{4}) |)$

Step 2.2.2

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\ln (| \frac{\sqrt{3}}{2} |) - \ln (| \frac{\sqrt{2}}{2} |)$

Step 2.2.3

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

$\ln (\frac{| \frac{\sqrt{3}}{2} |}{| \frac{\sqrt{2}}{2} |})$

Step 2.3

Simplify.

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Step 2.3.1

$\frac{\sqrt{3}}{2}$ is approximately $0.8660254$ which is positive so remove the absolute value

$\ln (\frac{\frac{\sqrt{3}}{2}}{| \frac{\sqrt{2}}{2} |})$

Step 2.3.2

$\frac{\sqrt{2}}{2}$ is approximately $0.70710678$ which is positive so remove the absolute value

$\ln (\frac{\frac{\sqrt{3}}{2}}{\frac{\sqrt{2}}{2}})$

Step 2.3.3

Multiply the numerator by the reciprocal of the denominator.

$\ln (\frac{\sqrt{3}}{2} \cdot \frac{2}{\sqrt{2}})$

Step 2.3.4

Cancel the common factor of $2$ .

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Step 2.3.4.1

Cancel the common factor.

$\ln (\frac{\sqrt{3}}{2} \cdot \frac{2}{\sqrt{2}})$

Step 2.3.4.2

Rewrite the expression.

$\ln (\sqrt{3} \frac{1}{\sqrt{2}})$

Step 2.3.5

Combine $\sqrt{3}$ and $\frac{1}{\sqrt{2}}$ .

$\ln (\frac{\sqrt{3}}{\sqrt{2}})$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\ln (\frac{\sqrt{3}}{\sqrt{2}})$

Decimal Form:

$0.20273255 \dots$