Calculus Examples

$\int_{- 2}^{- \frac{2 \sqrt{3}}{3}} \frac{9}{x \sqrt{4 x^{2} - 4}} d x$

Step 1

Since

$9$ is constant with respect to

$x$ , move

$9$ out of the integral.

$9 \int_{- 2}^{- \frac{2 \sqrt{3}}{3}} \frac{1}{x \sqrt{4 x^{2} - 4}} d x$

Step 2

Simplify with factoring out.

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Step 2.1

Rewrite

$4 x^{2}$ as

${(2 x)}^{2}$ .

$9 \int_{- 2}^{- \frac{2 \sqrt{3}}{3}} \frac{1}{x \sqrt{{(2 x)}^{2} - 4}} d x$

Step 2.2

Rewrite

$4$ as

$2^{2}$ .

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

Step 3

The integral of

$\frac{1}{x \sqrt{{(2 x)}^{2} - 2^{2}}}$ with respect to

$x$ is

$\frac{1}{2} arcsec (| \frac{2 x}{2} |)$

$9 (\frac{1}{2} arcsec (| \frac{2 x}{2} |)]_{- 2}^{- \frac{2 \sqrt{3}}{3}})$

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

$2$ .

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Step 4.1.1.1

Cancel the common factor.

$9 (\frac{1}{2} arcsec (| \frac{2 x}{2} |)]_{- 2}^{- \frac{2 \sqrt{3}}{3}})$

Step 4.1.1.2

Divide

$x$ by

$1$ .

$9 (\frac{1}{2} arcsec (| x |)]_{- 2}^{- \frac{2 \sqrt{3}}{3}})$

Step 4.1.2

Combine

$\frac{1}{2}$ and

$arcsec (| x |)$ .

$9 (\frac{arcsec (| x |)}{2}]_{- 2}^{- \frac{2 \sqrt{3}}{3}})$

Step 4.2

Evaluate

$\frac{arcsec (| x |)}{2}$ at

$- \frac{2 \sqrt{3}}{3}$ and at

$- 2$ .

$9 (\frac{arcsec (| - \frac{2 \sqrt{3}}{3} |)}{2} - \frac{arcsec (| - 2 |)}{2})$

Step 5

Simplify.

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Step 5.1

Combine the numerators over the common denominator.

$9 \frac{arcsec (| - \frac{2 \sqrt{3}}{3} |) - arcsec (| - 2 |)}{2}$

Step 5.2

Simplify each term.

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Step 5.2.1

$- \frac{2 \sqrt{3}}{3}$ is approximately

$- 1.15470053$ which is negative so negate

$- \frac{2 \sqrt{3}}{3}$ and remove the absolute value

$9 \frac{arcsec (\frac{2 \sqrt{3}}{3}) - arcsec (| - 2 |)}{2}$

Step 5.2.2

The exact value of

$arcsec (\frac{2 \sqrt{3}}{3})$ is

$\frac{π}{6}$ .

$9 \frac{\frac{π}{6} - arcsec (| - 2 |)}{2}$

Step 5.2.3

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

$- 2$ and

$0$ is

$2$ .

$9 \frac{\frac{π}{6} - arcsec (2)}{2}$

Step 5.2.4

The exact value of

$arcsec (2)$ is

$\frac{π}{3}$ .

$9 \frac{\frac{π}{6} - \frac{π}{3}}{2}$

Step 5.3

To write

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

$\frac{2}{2}$ .

$9 \frac{\frac{π}{6} - \frac{π}{3} \cdot \frac{2}{2}}{2}$

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

$\frac{2}{2}$ .

$9 \frac{\frac{π}{6} - \frac{π \cdot 2}{3 \cdot 2}}{2}$

Step 5.4.2

Multiply

$3$ by

$2$ .

$9 \frac{\frac{π}{6} - \frac{π \cdot 2}{6}}{2}$

Step 5.5

Combine the numerators over the common denominator.

$9 \frac{\frac{π - π \cdot 2}{6}}{2}$

Step 5.6

Simplify the numerator.

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Step 5.6.1

Multiply

$2$ by

$- 1$ .

$9 \frac{\frac{π - 2 π}{6}}{2}$

Step 5.6.2

Subtract

$2 π$ from

$π$ .

$9 \frac{\frac{- π}{6}}{2}$

Step 5.7

Move the negative in front of the fraction.

$9 \frac{- \frac{π}{6}}{2}$

Step 5.8

Multiply the numerator by the reciprocal of the denominator.

$9 (- \frac{π}{6} \cdot \frac{1}{2})$

Step 5.9

Multiply

$- \frac{π}{6} \cdot \frac{1}{2}$ .

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Step 5.9.1

Multiply

$\frac{1}{2}$ by

$\frac{π}{6}$ .

$9 (- \frac{π}{2 \cdot 6})$

Step 5.9.2

Multiply

$2$ by

$6$ .

$9 (- \frac{π}{12})$

Step 5.10

Cancel the common factor of

$3$ .

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Step 5.10.1

Move the leading negative in

$- \frac{π}{12}$ into the numerator.

$9 \frac{- π}{12}$

Step 5.10.2

Factor

$3$ out of

$9$ .

$3 (3) \frac{- π}{12}$

Step 5.10.3

Factor

$3$ out of

$12$ .

$3 \cdot 3 \frac{- π}{3 \cdot 4}$

Step 5.10.4

Cancel the common factor.

$3 \cdot 3 \frac{- π}{3 \cdot 4}$

Step 5.10.5

Rewrite the expression.

$3 \frac{- π}{4}$

Step 5.11

Combine

$3$ and

$\frac{- π}{4}$ .

$\frac{3 (- π)}{4}$

Step 5.12

Multiply

$- 1$ by

$3$ .

$\frac{- 3 π}{4}$

Step 5.13

Move the negative in front of the fraction.

$- \frac{3 π}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{3 π}{4}$

Decimal Form:

$- 2.35619449 \dots$

Step 7