Calculus Examples

$\int_{\frac{π}{4}}^{\frac{π}{2}} \frac{\cos (x)}{\sin (x)} d x$

Step 1

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

$\int_{\frac{π}{4}}^{\frac{π}{2}} \cot (x) d x$

Step 2

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

$\ln (| \sin (x) |)]_{\frac{π}{4}}^{\frac{π}{2}}$

Step 3

Simplify the answer.

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Step 3.1

Evaluate $\ln (| \sin (x) |)$ at $\frac{π}{2}$ and at $\frac{π}{4}$ .

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

Step 3.2

Simplify.

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Step 3.2.1

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

$\ln (| 1 |) - \ln (| \sin (\frac{π}{4}) |)$

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.3

Simplify.

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.3.4

Multiply $\frac{2}{\sqrt{2}}$ by $1$ .

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.34657359 \dots$