Calculus Examples

$\int_{\frac{π}{4}}^{\frac{π}{2}} \cot^{3} (x) d x$

Step 1

Apply the reduction formula.

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Substitute and simplify.

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Step 3.1.1

Evaluate $- \frac{\cot^{2} (x)}{2}$ at $\frac{π}{2}$ and at $\frac{π}{4}$ .

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

Step 3.1.2

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

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

Step 3.1.3

Simplify.

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Step 3.1.3.1

To write $- (\ln (| \sin (\frac{π}{2}) |) - \ln (| \sin (\frac{π}{4}) |))$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.1.3.2

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

$\frac{\cot^{2} (\frac{π}{4})}{2} - \frac{\cot^{2} (\frac{π}{2})}{2} + \frac{- (\ln (| \sin (\frac{π}{2}) |) - \ln (| \sin (\frac{π}{4}) |)) \cdot 2}{2}$

Step 3.1.3.3

Combine the numerators over the common denominator.

$\frac{\cot^{2} (\frac{π}{4})}{2} + \frac{- \cot^{2} (\frac{π}{2}) - (\ln (| \sin (\frac{π}{2}) |) - \ln (| \sin (\frac{π}{4}) |)) \cdot 2}{2}$

Step 3.1.3.4

Multiply $2$ by $- 1$ .

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

Step 3.2

Simplify.

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Step 3.2.1

The exact value of $\cot (\frac{π}{4})$ is $1$ .

$\frac{1^{2}}{2} + \frac{- \cot^{2} (\frac{π}{2}) - 2 (\ln (| \sin (\frac{π}{2}) |) - \ln (| \sin (\frac{π}{4}) |))}{2}$

Step 3.2.2

The exact value of $\cot (\frac{π}{2})$ is $0$ .

$\frac{1^{2}}{2} + \frac{- 0^{2} - 2 (\ln (| \sin (\frac{π}{2}) |) - \ln (| \sin (\frac{π}{4}) |))}{2}$

Step 3.2.3

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

$\frac{1^{2}}{2} + \frac{- 0^{2} - 2 (\ln (| 1 |) - \ln (| \sin (\frac{π}{4}) |))}{2}$

Step 3.2.4

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

$\frac{1^{2}}{2} + \frac{- 0^{2} - 2 (\ln (| 1 |) - \ln (| \frac{\sqrt{2}}{2} |))}{2}$

Step 3.2.5

One to any power is one.

$\frac{1}{2} + \frac{- 0^{2} - 2 (\ln (| 1 |) - \ln (| \frac{\sqrt{2}}{2} |))}{2}$

Step 3.2.6

Raising $0$ to any positive power yields $0$ .

$\frac{1}{2} + \frac{- 0 - 2 (\ln (| 1 |) - \ln (| \frac{\sqrt{2}}{2} |))}{2}$

Step 3.2.7

Multiply $- 1$ by $0$ .

$\frac{1}{2} + \frac{0 - 2 (\ln (| 1 |) - \ln (| \frac{\sqrt{2}}{2} |))}{2}$

Step 3.2.8

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

$\frac{1}{2} + \frac{0 - 2 \ln (\frac{| 1 |}{| \frac{\sqrt{2}}{2} |})}{2}$

Step 3.2.9

Subtract $2 \ln (\frac{| 1 |}{| \frac{\sqrt{2}}{2} |})$ from $0$ .

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

Step 3.2.10

Cancel the common factor of $- 2$ and $2$ .

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Step 3.2.10.1

Factor $2$ out of $- 2 \ln (\frac{| 1 |}{| \frac{\sqrt{2}}{2} |})$ .

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

Step 3.2.10.2

Cancel the common factors.

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Step 3.2.10.2.1

Factor $2$ out of $2$ .

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

Step 3.2.10.2.2

Cancel the common factor.

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

Step 3.2.10.2.3

Rewrite the expression.

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

Step 3.2.10.2.4

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

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

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 $1$ .

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

Step 3.3.2

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

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

Step 3.3.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.3.4

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

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.15342640 \dots$