Calculus Examples

$\int_{\frac{2 \sqrt{3}}{3}}^{2} \frac{12}{x \sqrt{9 x^{2} - 9}} d x$

Step 1

Since $12$ is constant with respect to $x$ , move $12$ out of the integral.

$12 \int_{\frac{2 \sqrt{3}}{3}}^{2} \frac{1}{x \sqrt{9 x^{2} - 9}} d x$

Step 2

Simplify with factoring out.

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Step 2.1

Rewrite $9 x^{2}$ as ${(3 x)}^{2}$ .

$12 \int_{\frac{2 \sqrt{3}}{3}}^{2} \frac{1}{x \sqrt{{(3 x)}^{2} - 9}} d x$

Step 2.2

Rewrite $9$ as $3^{2}$ .

$12 \int_{\frac{2 \sqrt{3}}{3}}^{2} \frac{1}{x \sqrt{{(3 x)}^{2} - 3^{2}}} d x$

Step 3

The integral of $\frac{1}{x \sqrt{{(3 x)}^{2} - 3^{2}}}$ with respect to $x$ is $\frac{1}{3} arcsec (| \frac{3 x}{3} |)$

$12 (\frac{1}{3} arcsec (| \frac{3 x}{3} |)]_{\frac{2 \sqrt{3}}{3}}^{2})$

Step 4

Simplify the answer.

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Step 4.1

Simplify.

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Step 4.1.1

Cancel the common factor of $3$ .

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Step 4.1.1.1

Cancel the common factor.

$12 (\frac{1}{3} arcsec (| \frac{3 x}{3} |)]_{\frac{2 \sqrt{3}}{3}}^{2})$

Step 4.1.1.2

Divide $x$ by $1$ .

$12 (\frac{1}{3} arcsec (| x |)]_{\frac{2 \sqrt{3}}{3}}^{2})$

Step 4.1.2

Combine $\frac{1}{3}$ and $arcsec (| x |)$ .

$12 (\frac{arcsec (| x |)}{3}]_{\frac{2 \sqrt{3}}{3}}^{2})$

Step 4.2

Evaluate $\frac{arcsec (| x |)}{3}$ at $2$ and at $\frac{2 \sqrt{3}}{3}$ .

$12 (\frac{arcsec (| 2 |)}{3} - \frac{arcsec (| \frac{2 \sqrt{3}}{3} |)}{3})$

Step 5

Simplify.

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Step 5.1

Combine the numerators over the common denominator.

$12 \frac{arcsec (| 2 |) - arcsec (| \frac{2 \sqrt{3}}{3} |)}{3}$

Step 5.2

Simplify each term.

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Step 5.2.1

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$12 \frac{arcsec (2) - arcsec (| \frac{2 \sqrt{3}}{3} |)}{3}$

Step 5.2.2

The exact value of $arcsec (2)$ is $\frac{π}{3}$ .

$12 \frac{\frac{π}{3} - arcsec (| \frac{2 \sqrt{3}}{3} |)}{3}$

Step 5.2.3

$\frac{2 \sqrt{3}}{3}$ is approximately $1.15470053$ which is positive so remove the absolute value

$12 \frac{\frac{π}{3} - arcsec (\frac{2 \sqrt{3}}{3})}{3}$

Step 5.2.4

The exact value of $arcsec (\frac{2 \sqrt{3}}{3})$ is $\frac{π}{6}$ .

$12 \frac{\frac{π}{3} - \frac{π}{6}}{3}$

Step 5.3

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

$12 \frac{\frac{π}{3} \cdot \frac{2}{2} - \frac{π}{6}}{3}$

Step 5.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.4.1

Multiply $\frac{π}{3}$ by $\frac{2}{2}$ .

$12 \frac{\frac{π \cdot 2}{3 \cdot 2} - \frac{π}{6}}{3}$

Step 5.4.2

Multiply $3$ by $2$ .

$12 \frac{\frac{π \cdot 2}{6} - \frac{π}{6}}{3}$

Step 5.5

Combine the numerators over the common denominator.

$12 \frac{\frac{π \cdot 2 - π}{6}}{3}$

Step 5.6

Simplify the numerator.

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Step 5.6.1

Move $2$ to the left of $π$ .

$12 \frac{\frac{2 \cdot π - π}{6}}{3}$

Step 5.6.2

Subtract $π$ from $2 π$ .

$12 \frac{\frac{π}{6}}{3}$

Step 5.7

Cancel the common factor of $3$ .

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Step 5.7.1

Factor $3$ out of $12$ .

$3 (4) \frac{\frac{π}{6}}{3}$

Step 5.7.2

Cancel the common factor.

$3 \cdot 4 \frac{\frac{π}{6}}{3}$

Step 5.7.3

Rewrite the expression.

$4 \frac{π}{6}$

Step 5.8

Combine $4$ and $\frac{π}{6}$ .

$\frac{4 π}{6}$

Step 5.9

Cancel the common factor of $4$ and $6$ .

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Step 5.9.1

Factor $2$ out of $4 π$ .

$\frac{2 (2 π)}{6}$

Step 5.9.2

Cancel the common factors.

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Step 5.9.2.1

Factor $2$ out of $6$ .

$\frac{2 (2 π)}{2 (3)}$

Step 5.9.2.2

Cancel the common factor.

$\frac{2 (2 π)}{2 \cdot 3}$

Step 5.9.2.3

Rewrite the expression.

$\frac{2 π}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{2 π}{3}$

Decimal Form:

$2.09439510 \dots$

Step 7