Calculus Examples

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

Step 1

This integral could not be completed using u-substitution. Mathway will use another method.

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

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

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

Step 3.2

Simplify.

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Step 3.2.1

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

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

Step 3.2.2

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

$\ln (| \csc (\frac{3 π}{4}) - \cot (\frac{3 π}{4}) |) - \ln (| \sqrt{2} - 1 \cdot 1 |)$

Step 3.2.3

Multiply $- 1$ by $1$ .

$\ln (| \csc (\frac{3 π}{4}) - \cot (\frac{3 π}{4}) |) - \ln (| \sqrt{2} - 1 |)$

Step 3.2.4

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

$\ln (\frac{| \csc (\frac{3 π}{4}) - \cot (\frac{3 π}{4}) |}{| \sqrt{2} - 1 |})$

Step 3.3

Simplify.

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Step 3.3.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cotangent is negative in the second quadrant.

$\ln (\frac{| \csc (\frac{3 π}{4}) - - \cot (\frac{π}{4}) |}{| \sqrt{2} - 1 |})$

Step 3.3.2

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

$\ln (\frac{| \csc (\frac{3 π}{4}) - (- 1 \cdot 1) |}{| \sqrt{2} - 1 |})$

Step 3.3.3

Multiply $- 1$ by $1$ .

$\ln (\frac{| \csc (\frac{3 π}{4}) - - 1 |}{| \sqrt{2} - 1 |})$

Step 3.3.4

Multiply $- 1$ by $- 1$ .

$\ln (\frac{| \csc (\frac{3 π}{4}) + 1 |}{| \sqrt{2} - 1 |})$

Step 3.3.5

Simplify the numerator.

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Step 3.3.5.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\ln (\frac{| \csc (\frac{π}{4}) + 1 |}{| \sqrt{2} - 1 |})$

Step 3.3.5.2

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

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

Step 3.3.5.3

$\sqrt{2} + 1$ is approximately $2.41421356$ which is positive so remove the absolute value

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

Step 3.3.6

$\sqrt{2} - 1$ is approximately $0.41421356$ which is positive so remove the absolute value

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Expand the denominator using the FOIL method.

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

Step 4.4

Simplify.

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

Step 4.5

Divide $(\sqrt{2} + 1) (\sqrt{2} + 1)$ by $1$ .

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

Step 4.6

Rewrite $\ln ((\sqrt{2} + 1) (\sqrt{2} + 1))$ as $\ln (\sqrt{2} + 1) + \ln (\sqrt{2} + 1)$ .

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

Step 4.7

Add $\ln (\sqrt{2} + 1)$ and $\ln (\sqrt{2} + 1)$ .

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$1.76274717 \dots$