Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

$\int_{\frac{π}{4}}^{\frac{π}{2}} 2 d x + \int_{\frac{π}{4}}^{\frac{π}{2}} - \csc^{2} (x) d x$

Step 3

Apply the constant rule.

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

Step 4

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 5

Since the derivative of $- \cot (x)$ is $\csc^{2} (x)$ , the integral of $\csc^{2} (x)$ is $- \cot (x)$ .

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

Step 6

Simplify the answer.

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Step 6.1

Substitute and simplify.

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Step 6.1.1

Evaluate $2 x$ at $\frac{π}{2}$ and at $\frac{π}{4}$ .

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

Step 6.1.2

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

$2 \frac{π}{2} - 2 \frac{π}{4} - (- \cot (\frac{π}{2}) + \cot (\frac{π}{4}))$

Step 6.1.3

Simplify.

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Step 6.1.3.1

Combine $2$ and $\frac{π}{2}$ .

$\frac{2 π}{2} - 2 \frac{π}{4} - (- \cot (\frac{π}{2}) + \cot (\frac{π}{4}))$

Step 6.1.3.2

Cancel the common factor of $2$ .

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Step 6.1.3.2.1

Cancel the common factor.

$\frac{2 π}{2} - 2 \frac{π}{4} - (- \cot (\frac{π}{2}) + \cot (\frac{π}{4}))$

Step 6.1.3.2.2

Divide $π$ by $1$ .

$π - 2 \frac{π}{4} - (- \cot (\frac{π}{2}) + \cot (\frac{π}{4}))$

Step 6.1.3.3

Combine $- 2$ and $\frac{π}{4}$ .

$π + \frac{- 2 π}{4} - (- \cot (\frac{π}{2}) + \cot (\frac{π}{4}))$

Step 6.1.3.4

Cancel the common factor of $- 2$ and $4$ .

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Step 6.1.3.4.1

Factor $2$ out of $- 2 π$ .

$π + \frac{2 (- π)}{4} - (- \cot (\frac{π}{2}) + \cot (\frac{π}{4}))$

Step 6.1.3.4.2

Cancel the common factors.

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Step 6.1.3.4.2.1

Factor $2$ out of $4$ .

$π + \frac{2 (- π)}{2 (2)} - (- \cot (\frac{π}{2}) + \cot (\frac{π}{4}))$

Step 6.1.3.4.2.2

Cancel the common factor.

$π + \frac{2 (- π)}{2 \cdot 2} - (- \cot (\frac{π}{2}) + \cot (\frac{π}{4}))$

Step 6.1.3.4.2.3

Rewrite the expression.

$π + \frac{- π}{2} - (- \cot (\frac{π}{2}) + \cot (\frac{π}{4}))$

Step 6.1.3.5

Move the negative in front of the fraction.

$π - \frac{π}{2} - (- \cot (\frac{π}{2}) + \cot (\frac{π}{4}))$

Step 6.1.3.6

To write $π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$π \cdot \frac{2}{2} - \frac{π}{2} - (- \cot (\frac{π}{2}) + \cot (\frac{π}{4}))$

Step 6.1.3.7

Combine $π$ and $\frac{2}{2}$ .

$\frac{π \cdot 2}{2} - \frac{π}{2} - (- \cot (\frac{π}{2}) + \cot (\frac{π}{4}))$

Step 6.1.3.8

Combine the numerators over the common denominator.

$\frac{π \cdot 2 - π}{2} - (- \cot (\frac{π}{2}) + \cot (\frac{π}{4}))$

Step 6.1.3.9

Move $2$ to the left of $π$ .

$\frac{2 \cdot π - π}{2} - (- \cot (\frac{π}{2}) + \cot (\frac{π}{4}))$

Step 6.1.3.10

Subtract $π$ from $2 π$ .

$\frac{π}{2} - (- \cot (\frac{π}{2}) + \cot (\frac{π}{4}))$

Step 6.2

Simplify.

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Step 6.2.1

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

$\frac{π}{2} - (- 0 + \cot (\frac{π}{4}))$

Step 6.2.2

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

$\frac{π}{2} - (- 0 + 1)$

Step 6.2.3

Multiply $- 1$ by $0$ .

$\frac{π}{2} - (0 + 1)$

Step 6.2.4

Add $0$ and $1$ .

$\frac{π}{2} - 1 \cdot 1$

Step 6.2.5

Multiply $- 1$ by $1$ .

$\frac{π}{2} - 1$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{π}{2} - 1$

Decimal Form:

$0.57079632 \dots$