Calculus Examples

$\int_{\frac{π}{6}}^{\frac{π}{2}} 7 \cot^{2} (x) d x$

Step 1

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$7 \int_{\frac{π}{6}}^{\frac{π}{2}} \cot^{2} (x) d x$

Step 2

Using the Pythagorean Identity, rewrite $\cot^{2} (x)$ as $- 1 + \csc^{2} (x)$ .

$7 \int_{\frac{π}{6}}^{\frac{π}{2}} - 1 + \csc^{2} (x) d x$

Step 3

Split the single integral into multiple integrals.

$7 (\int_{\frac{π}{6}}^{\frac{π}{2}} - 1 d x + \int_{\frac{π}{6}}^{\frac{π}{2}} \csc^{2} (x) d x)$

Step 4

Apply the constant rule.

$7 (- x]_{\frac{π}{6}}^{\frac{π}{2}} + \int_{\frac{π}{6}}^{\frac{π}{2}} \csc^{2} (x) d x)$

Step 5

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

$7 (- x]_{\frac{π}{6}}^{\frac{π}{2}} + - \cot (x)]_{\frac{π}{6}}^{\frac{π}{2}})$

Step 6

Simplify the answer.

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Step 6.1

Combine $- x]_{\frac{π}{6}}^{\frac{π}{2}}$ and $- \cot (x)]_{\frac{π}{6}}^{\frac{π}{2}}$ .

$7 (- x - \cot (x)]_{\frac{π}{6}}^{\frac{π}{2}})$

Step 6.2

Evaluate $- x - \cot (x)$ at $\frac{π}{2}$ and at $\frac{π}{6}$ .

$7 (- \frac{π}{2} - \cot (\frac{π}{2}) - (- \frac{π}{6} - \cot (\frac{π}{6})))$

Step 6.3

Simplify.

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Step 6.3.1

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

$7 (- \frac{π}{2} - 0 - (- \frac{π}{6} - \cot (\frac{π}{6})))$

Step 6.3.2

The exact value of $\cot (\frac{π}{6})$ is $\sqrt{3}$ .

$7 (- \frac{π}{2} - 0 - (- \frac{π}{6} - \sqrt{3}))$

Step 6.3.3

Multiply $- 1$ by $0$ .

$7 (- \frac{π}{2} + 0 - (- \frac{π}{6} - \sqrt{3}))$

Step 6.3.4

Add $- \frac{π}{2}$ and $0$ .

$7 (- \frac{π}{2} - (- \frac{π}{6} - \sqrt{3}))$

Step 6.3.5

To write $- (- \frac{π}{6} - \sqrt{3})$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$7 (- \frac{π}{2} - (- \frac{π}{6} - \sqrt{3}) \cdot \frac{2}{2})$

Step 6.3.6

Combine $- (- \frac{π}{6} - \sqrt{3})$ and $\frac{2}{2}$ .

$7 (- \frac{π}{2} + \frac{- (- \frac{π}{6} - \sqrt{3}) \cdot 2}{2})$

Step 6.3.7

Combine the numerators over the common denominator.

$7 \frac{- π - (- \frac{π}{6} - \sqrt{3}) \cdot 2}{2}$

Step 6.3.8

Multiply $2$ by $- 1$ .

$7 \frac{- π - 2 (- \frac{π}{6} - \sqrt{3})}{2}$

Step 6.3.9

Combine $7$ and $\frac{- π - 2 (- \frac{π}{6} - \sqrt{3})}{2}$ .

$\frac{7 (- π - 2 (- \frac{π}{6} - \sqrt{3}))}{2}$

Step 6.3.10

Factor $- 1$ out of $- π$ .

$\frac{7 (- (π) - 2 (- \frac{π}{6} - \sqrt{3}))}{2}$

Step 6.3.11

Factor $- 1$ out of $- 2 (- \frac{π}{6} - \sqrt{3})$ .

$\frac{7 (- (π) - (2 (- \frac{π}{6} - \sqrt{3})))}{2}$

Step 6.3.12

Factor $- 1$ out of $- (π) - (2 (- \frac{π}{6} - \sqrt{3}))$ .

$\frac{7 (- (π + 2 (- \frac{π}{6} - \sqrt{3})))}{2}$

Step 6.3.13

Rewrite $- (π + 2 (- \frac{π}{6} - \sqrt{3}))$ as $- 1 (π + 2 (- \frac{π}{6} - \sqrt{3}))$ .

$\frac{7 (- 1 (π + 2 (- \frac{π}{6} - \sqrt{3})))}{2}$

Step 6.3.14

Move the negative in front of the fraction.

$- \frac{7 (π + 2 (- \frac{π}{6} - \sqrt{3}))}{2}$

Step 6.4

Simplify.

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Step 6.4.1

Apply the distributive property.

$- \frac{7 (π + 2 (- \frac{π}{6}) + 2 (- \sqrt{3}))}{2}$

Step 6.4.2

Cancel the common factor of $2$ .

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Step 6.4.2.1

Move the leading negative in $- \frac{π}{6}$ into the numerator.

$- \frac{7 (π + 2 \frac{- π}{6} + 2 (- \sqrt{3}))}{2}$

Step 6.4.2.2

Factor $2$ out of $6$ .

$- \frac{7 (π + 2 \frac{- π}{2 (3)} + 2 (- \sqrt{3}))}{2}$

Step 6.4.2.3

Cancel the common factor.

$- \frac{7 (π + 2 \frac{- π}{2 \cdot 3} + 2 (- \sqrt{3}))}{2}$

Step 6.4.2.4

Rewrite the expression.

$- \frac{7 (π + \frac{- π}{3} + 2 (- \sqrt{3}))}{2}$

Step 6.4.3

Multiply $- 1$ by $2$ .

$- \frac{7 (π + \frac{- π}{3} - 2 \sqrt{3})}{2}$

Step 6.4.4

Move the negative in front of the fraction.

$- \frac{7 (π - \frac{π}{3} - 2 \sqrt{3})}{2}$

Step 6.4.5

To write $π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{7 (π \cdot \frac{3}{3} - \frac{π}{3} - 2 \sqrt{3})}{2}$

Step 6.4.6

Combine $π$ and $\frac{3}{3}$ .

$- \frac{7 (\frac{π \cdot 3}{3} - \frac{π}{3} - 2 \sqrt{3})}{2}$

Step 6.4.7

Combine the numerators over the common denominator.

$- \frac{7 (\frac{π \cdot 3 - π}{3} - 2 \sqrt{3})}{2}$

Step 6.4.8

Move $3$ to the left of $π$ .

$- \frac{7 (\frac{3 \cdot π - π}{3} - 2 \sqrt{3})}{2}$

Step 6.4.9

Subtract $π$ from $3 π$ .

$- \frac{7 (\frac{2 π}{3} - 2 \sqrt{3})}{2}$

Step 6.4.10

Apply the distributive property.

$- \frac{7 \frac{2 π}{3} + 7 (- 2 \sqrt{3})}{2}$

Step 6.4.11

Multiply $7 \frac{2 π}{3}$ .

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Step 6.4.11.1

Combine $7$ and $\frac{2 π}{3}$ .

$- \frac{\frac{7 (2 π)}{3} + 7 (- 2 \sqrt{3})}{2}$

Step 6.4.11.2

Multiply $2$ by $7$ .

$- \frac{\frac{14 π}{3} + 7 (- 2 \sqrt{3})}{2}$

Step 6.4.12

Multiply $- 2$ by $7$ .

$- \frac{\frac{14 π}{3} - 14 \sqrt{3}}{2}$

Step 6.4.13

Cancel the common factor of $\frac{14 π}{3} - 14 \sqrt{3}$ and $2$ .

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Step 6.4.13.1

Factor $2$ out of $\frac{14 π}{3}$ .

$- \frac{2 \frac{7 π}{3} - 14 \sqrt{3}}{2}$

Step 6.4.13.2

Factor $2$ out of $- 14 \sqrt{3}$ .

$- \frac{2 \frac{7 π}{3} + 2 (- 7 \sqrt{3})}{2}$

Step 6.4.13.3

Factor $2$ out of $2 \frac{7 π}{3} + 2 (- 7 \sqrt{3})$ .

$- \frac{2 (\frac{7 π}{3} - 7 \sqrt{3})}{2}$

Step 6.4.13.4

Cancel the common factors.

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Step 6.4.13.4.1

Factor $2$ out of $2$ .

$- \frac{2 (\frac{7 π}{3} - 7 \sqrt{3})}{2 (1)}$

Step 6.4.13.4.2

Cancel the common factor.

$- \frac{2 (\frac{7 π}{3} - 7 \sqrt{3})}{2 \cdot 1}$

Step 6.4.13.4.3

Rewrite the expression.

$- \frac{\frac{7 π}{3} - 7 \sqrt{3}}{1}$

Step 6.4.13.4.4

Divide $\frac{7 π}{3} - 7 \sqrt{3}$ by $1$ .

$- (\frac{7 π}{3} - 7 \sqrt{3})$

Step 6.4.14

Apply the distributive property.

$- \frac{7 π}{3} - (- 7 \sqrt{3})$

Step 6.4.15

Multiply $- 7$ by $- 1$ .

$- \frac{7 π}{3} + 7 \sqrt{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \frac{7 π}{3} + 7 \sqrt{3}$

Decimal Form:

$4.79397279 \dots$