Calculus Examples

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

Step 1

Let $x = 2 \sec (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = 2 \sec (t) \tan (t) d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $2 \sec (t) \tan (t)$ is positive.

$\int \frac{{(2 \sec (t))}^{3}}{\sqrt{{(2 \sec (t))}^{2} - 4}} (2 \sec (t) \tan (t)) d t$

Step 2

Simplify terms.

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Step 2.1

Simplify $\sqrt{{(2 \sec (t))}^{2} - 4}$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the product rule to $2 \sec (t)$ .

$\int \frac{{(2 \sec (t))}^{3}}{\sqrt{2^{2} \sec^{2} (t) - 4}} (2 \sec (t) \tan (t)) d t$

Step 2.1.1.2

Raise $2$ to the power of $2$ .

$\int \frac{{(2 \sec (t))}^{3}}{\sqrt{4 \sec^{2} (t) - 4}} (2 \sec (t) \tan (t)) d t$

Step 2.1.2

Factor $4$ out of $4 \sec^{2} (t)$ .

$\int \frac{{(2 \sec (t))}^{3}}{\sqrt{4 (\sec^{2} (t)) - 4}} (2 \sec (t) \tan (t)) d t$

Step 2.1.3

Factor $4$ out of $- 4$ .

$\int \frac{{(2 \sec (t))}^{3}}{\sqrt{4 \sec^{2} (t) + 4 \cdot - 1}} (2 \sec (t) \tan (t)) d t$

Step 2.1.4

Factor $4$ out of $4 \sec^{2} (t) + 4 \cdot - 1$ .

$\int \frac{{(2 \sec (t))}^{3}}{\sqrt{4 (\sec^{2} (t) - 1)}} (2 \sec (t) \tan (t)) d t$

Step 2.1.5

Apply pythagorean identity.

$\int \frac{{(2 \sec (t))}^{3}}{\sqrt{4 \tan^{2} (t)}} (2 \sec (t) \tan (t)) d t$

Step 2.1.6

Rewrite $4 \tan^{2} (t)$ as ${(2 \tan (t))}^{2}$ .

$\int \frac{{(2 \sec (t))}^{3}}{\sqrt{{(2 \tan (t))}^{2}}} (2 \sec (t) \tan (t)) d t$

Step 2.1.7

Pull terms out from under the radical, assuming positive real numbers.

$\int \frac{{(2 \sec (t))}^{3}}{2 \tan (t)} (2 \sec (t) \tan (t)) d t$

Step 2.2

Reduce the expression by cancelling the common factors.

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Step 2.2.1

Cancel the common factor of $2 \tan (t)$ .

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Step 2.2.1.1

Factor $2 \tan (t)$ out of $2 \sec (t) \tan (t)$ .

$\int \frac{{(2 \sec (t))}^{3}}{2 \tan (t)} (2 \tan (t) (\sec (t))) d t$

Step 2.2.1.2

Cancel the common factor.

$\int \frac{{(2 \sec (t))}^{3}}{2 \tan (t)} (2 \tan (t) \sec (t)) d t$

Step 2.2.1.3

Rewrite the expression.

$\int {(2 \sec (t))}^{3} \sec (t) d t$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Factor $2$ out of $2 \sec (t)$ .

$\int {(2 (\sec (t)))}^{3} \sec (t) d t$

Step 2.2.2.2

Apply the product rule to $2 (\sec (t))$ .

$\int 2^{3} \sec^{3} (t) \sec (t) d t$

Step 2.2.2.3

Raise $2$ to the power of $3$ .

$\int 8 \sec^{3} (t) \sec (t) d t$

Step 2.2.2.4

Raise $\sec (t)$ to the power of $1$ .

$\int 8 (\sec^{1} (t) \sec^{3} (t)) d t$

Step 2.2.2.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int 8 {\sec (t)}^{1 + 3} d t$

Step 2.2.2.6

Add $1$ and $3$ .

$\int 8 \sec^{4} (t) d t$

Step 3

Since $8$ is constant with respect to $t$ , move $8$ out of the integral.

$8 \int \sec^{4} (t) d t$

Step 4

Simplify the expression.

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Step 4.1

Rewrite $4$ as $2$ plus $2$

$8 \int {\sec (t)}^{2 + 2} d t$

Step 4.2

Rewrite ${\sec (t)}^{2 + 2}$ as $\sec^{2} (t) \sec^{2} (t)$ .

$8 \int \sec^{2} (t) \sec^{2} (t) d t$

Step 5

Using the Pythagorean Identity, rewrite $\sec^{2} (t)$ as $1 + \tan^{2} (t)$ .

$8 \int (1 + \tan^{2} (t)) \sec^{2} (t) d t$

Step 6

Let $u = \tan (t)$ . Then $d u = \sec^{2} (t) d t$ , so $\frac{1}{\sec^{2} (t)} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.1

Let $u = \tan (t)$ . Find $\frac{d u}{d t}$ .

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Step 6.1.1

Differentiate $\tan (t)$ .

$\frac{d}{d t} [\tan (t)]$

Step 6.1.2

The derivative of $\tan (t)$ with respect to $t$ is $\sec^{2} (t)$ .

$\sec^{2} (t)$

Step 6.2

Rewrite the problem using $u$ and $d u$ .

$8 \int 1 + u^{2} d u$

Step 7

Split the single integral into multiple integrals.

$8 (\int d u + \int u^{2} d u)$

Step 8

Apply the constant rule.

$8 (u + C + \int u^{2} d u)$

Step 9

By the Power Rule, the integral of $u^{2}$ with respect to $u$ is $\frac{1}{3} u^{3}$ .

$8 (u + C + \frac{1}{3} u^{3} + C)$

Step 10

Simplify.

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Step 10.1

Combine $\frac{1}{3}$ and $u^{3}$ .

$8 (u + C + \frac{u^{3}}{3} + C)$

Step 10.2

Simplify.

$8 (u + \frac{1}{3} u^{3}) + C$

Step 11

Substitute back in for each integration substitution variable.

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Step 11.1

Replace all occurrences of $u$ with $\tan (t)$ .

$8 (\tan (t) + \frac{1}{3} \tan^{3} (t)) + C$

Step 11.2

Replace all occurrences of $t$ with $arcsec (\frac{x}{2})$ .

$8 (\tan (arcsec (\frac{x}{2})) + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{2}))) + C$

Step 12

Simplify.

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Step 12.1

Simplify each term.

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Step 12.1.1

Draw a triangle in the plane with vertices $(1, \sqrt{{(\frac{x}{2})}^{2} - 1^{2}})$ , $(1, 0)$ , and the origin. Then $arcsec (\frac{x}{2})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, \sqrt{{(\frac{x}{2})}^{2} - 1^{2}})$ . Therefore, $\tan (arcsec (\frac{x}{2}))$ is $\sqrt{{(\frac{x}{2})}^{2} - 1}$ .

$8 (\sqrt{{(\frac{x}{2})}^{2} - 1} + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{2}))) + C$

Step 12.1.2

Rewrite $1$ as $1^{2}$ .

$8 (\sqrt{{(\frac{x}{2})}^{2} - 1^{2}} + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{2}))) + C$

Step 12.1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \frac{x}{2}$ and $b = 1$ .

$8 (\sqrt{(\frac{x}{2} + 1) (\frac{x}{2} - 1)} + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{2}))) + C$

Step 12.1.4

Write $1$ as a fraction with a common denominator.

$8 (\sqrt{(\frac{x}{2} + \frac{2}{2}) (\frac{x}{2} - 1)} + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{2}))) + C$

Step 12.1.5

Combine the numerators over the common denominator.

$8 (\sqrt{\frac{x + 2}{2} (\frac{x}{2} - 1)} + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{2}))) + C$

Step 12.1.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$8 (\sqrt{\frac{x + 2}{2} (\frac{x}{2} - 1 \cdot \frac{2}{2})} + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{2}))) + C$

Step 12.1.7

Combine $- 1$ and $\frac{2}{2}$ .

$8 (\sqrt{\frac{x + 2}{2} (\frac{x}{2} + \frac{- 1 \cdot 2}{2})} + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{2}))) + C$

Step 12.1.8

Combine the numerators over the common denominator.

$8 (\sqrt{\frac{x + 2}{2} \cdot \frac{x - 1 \cdot 2}{2}} + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{2}))) + C$

Step 12.1.9

Multiply $- 1$ by $2$ .

$8 (\sqrt{\frac{x + 2}{2} \cdot \frac{x - 2}{2}} + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{2}))) + C$

Step 12.1.10

Multiply $\frac{x + 2}{2}$ by $\frac{x - 2}{2}$ .

$8 (\sqrt{\frac{(x + 2) (x - 2)}{2 \cdot 2}} + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{2}))) + C$

Step 12.1.11

Multiply $2$ by $2$ .

$8 (\sqrt{\frac{(x + 2) (x - 2)}{4}} + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{2}))) + C$

Step 12.1.12

Rewrite $\frac{(x + 2) (x - 2)}{4}$ as ${(\frac{1}{2})}^{2} ((x + 2) (x - 2))$ .

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Step 12.1.12.1

Factor the perfect power $1^{2}$ out of $(x + 2) (x - 2)$ .

$8 (\sqrt{\frac{1^{2} ((x + 2) (x - 2))}{4}} + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{2}))) + C$

Step 12.1.12.2

Factor the perfect power $2^{2}$ out of $4$ .

$8 (\sqrt{\frac{1^{2} ((x + 2) (x - 2))}{2^{2} \cdot 1}} + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{2}))) + C$

Step 12.1.12.3

Rearrange the fraction $\frac{1^{2} ((x + 2) (x - 2))}{2^{2} \cdot 1}$ .

$8 (\sqrt{{(\frac{1}{2})}^{2} ((x + 2) (x - 2))} + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{2}))) + C$

Step 12.1.13

Pull terms out from under the radical.

$8 (\frac{1}{2} \sqrt{(x + 2) (x - 2)} + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{2}))) + C$

Step 12.1.14

Combine $\frac{1}{2}$ and $\sqrt{(x + 2) (x - 2)}$ .

$8 (\frac{\sqrt{(x + 2) (x - 2)}}{2} + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{2}))) + C$

Step 12.2

Combine $\frac{1}{3}$ and $\tan^{3} (arcsec (\frac{x}{2}))$ .

$8 (\frac{\sqrt{(x + 2) (x - 2)}}{2} + \frac{\tan^{3} (arcsec (\frac{x}{2}))}{3}) + C$

Step 12.3

Apply the distributive property.

$8 \frac{\sqrt{(x + 2) (x - 2)}}{2} + 8 \frac{\tan^{3} (arcsec (\frac{x}{2}))}{3} + C$

Step 12.4

Cancel the common factor of $2$ .

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Step 12.4.1

Factor $2$ out of $8$ .

$2 (4) \frac{\sqrt{(x + 2) (x - 2)}}{2} + 8 \frac{\tan^{3} (arcsec (\frac{x}{2}))}{3} + C$

Step 12.4.2

Cancel the common factor.

$2 \cdot 4 \frac{\sqrt{(x + 2) (x - 2)}}{2} + 8 \frac{\tan^{3} (arcsec (\frac{x}{2}))}{3} + C$

Step 12.4.3

Rewrite the expression.

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{\tan^{3} (arcsec (\frac{x}{2}))}{3} + C$

Step 12.5

Simplify the numerator.

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Step 12.5.1

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{{\sqrt{{(\frac{x}{2})}^{2} - 1}}^{3}}{3} + C$

Step 12.5.2

Rewrite $1$ as $1^{2}$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{{\sqrt{{(\frac{x}{2})}^{2} - 1^{2}}}^{3}}{3} + C$

Step 12.5.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \frac{x}{2}$ and $b = 1$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{{\sqrt{(\frac{x}{2} + 1) (\frac{x}{2} - 1)}}^{3}}{3} + C$

Step 12.5.4

Write $1$ as a fraction with a common denominator.

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{{\sqrt{(\frac{x}{2} + \frac{2}{2}) (\frac{x}{2} - 1)}}^{3}}{3} + C$

Step 12.5.5

Combine the numerators over the common denominator.

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{{\sqrt{\frac{x + 2}{2} (\frac{x}{2} - 1)}}^{3}}{3} + C$

Step 12.5.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{{\sqrt{\frac{x + 2}{2} (\frac{x}{2} - 1 \cdot \frac{2}{2})}}^{3}}{3} + C$

Step 12.5.7

Combine $- 1$ and $\frac{2}{2}$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{{\sqrt{\frac{x + 2}{2} (\frac{x}{2} + \frac{- 1 \cdot 2}{2})}}^{3}}{3} + C$

Step 12.5.8

Combine the numerators over the common denominator.

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{{\sqrt{\frac{x + 2}{2} \cdot \frac{x - 1 \cdot 2}{2}}}^{3}}{3} + C$

Step 12.5.9

Multiply $- 1$ by $2$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{{\sqrt{\frac{x + 2}{2} \cdot \frac{x - 2}{2}}}^{3}}{3} + C$

Step 12.5.10

Multiply $\frac{x + 2}{2}$ by $\frac{x - 2}{2}$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{{\sqrt{\frac{(x + 2) (x - 2)}{2 \cdot 2}}}^{3}}{3} + C$

Step 12.5.11

Multiply $2$ by $2$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{{\sqrt{\frac{(x + 2) (x - 2)}{4}}}^{3}}{3} + C$

Step 12.5.12

Rewrite $\frac{(x + 2) (x - 2)}{4}$ as ${(\frac{1}{2})}^{2} ((x + 2) (x - 2))$ .

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Step 12.5.12.1

Factor the perfect power $1^{2}$ out of $(x + 2) (x - 2)$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{{\sqrt{\frac{1^{2} ((x + 2) (x - 2))}{4}}}^{3}}{3} + C$

Step 12.5.12.2

Factor the perfect power $2^{2}$ out of $4$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{{\sqrt{\frac{1^{2} ((x + 2) (x - 2))}{2^{2} \cdot 1}}}^{3}}{3} + C$

Step 12.5.12.3

Rearrange the fraction $\frac{1^{2} ((x + 2) (x - 2))}{2^{2} \cdot 1}$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{{\sqrt{{(\frac{1}{2})}^{2} ((x + 2) (x - 2))}}^{3}}{3} + C$

Step 12.5.13

Pull terms out from under the radical.

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{{(\frac{1}{2} \sqrt{(x + 2) (x - 2)})}^{3}}{3} + C$

Step 12.5.14

Combine $\frac{1}{2}$ and $\sqrt{(x + 2) (x - 2)}$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{{(\frac{\sqrt{(x + 2) (x - 2)}}{2})}^{3}}{3} + C$

Step 12.5.15

Apply the product rule to $\frac{\sqrt{(x + 2) (x - 2)}}{2}$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{\frac{{\sqrt{(x + 2) (x - 2)}}^{3}}{2^{3}}}{3} + C$

Step 12.5.16

Simplify the numerator.

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Step 12.5.16.1

Rewrite ${\sqrt{(x + 2) (x - 2)}}^{3}$ as $\sqrt{{((x + 2) (x - 2))}^{3}}$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{\frac{\sqrt{{((x + 2) (x - 2))}^{3}}}{2^{3}}}{3} + C$

Step 12.5.16.2

Apply the product rule to $(x + 2) (x - 2)$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{\frac{\sqrt{{(x + 2)}^{3} {(x - 2)}^{3}}}{2^{3}}}{3} + C$

Step 12.5.16.3

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

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Step 12.5.16.3.1

Factor out ${(x + 2)}^{2}$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{\frac{\sqrt{{(x + 2)}^{2} (x + 2) {(x - 2)}^{3}}}{2^{3}}}{3} + C$

Step 12.5.16.3.2

Factor out ${(x - 2)}^{2}$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{\frac{\sqrt{{(x + 2)}^{2} (x + 2) ({(x - 2)}^{2} (x - 2))}}{2^{3}}}{3} + C$

Step 12.5.16.3.3

Move $x + 2$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{\frac{\sqrt{{(x + 2)}^{2} {(x - 2)}^{2} (x + 2) (x - 2)}}{2^{3}}}{3} + C$

Step 12.5.16.3.4

Rewrite ${(x + 2)}^{2} {(x - 2)}^{2}$ as ${((x + 2) (x - 2))}^{2}$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{\frac{\sqrt{{((x + 2) (x - 2))}^{2} (x + 2) (x - 2)}}{2^{3}}}{3} + C$

Step 12.5.16.3.5

Add parentheses.

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{\frac{\sqrt{{((x + 2) (x - 2))}^{2} ((x + 2) (x - 2))}}{2^{3}}}{3} + C$

Step 12.5.16.4

Pull terms out from under the radical.

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{\frac{(x + 2) (x - 2) \sqrt{(x + 2) (x - 2)}}{2^{3}}}{3} + C$

Step 12.5.16.5

Expand $(x + 2) (x - 2)$ using the FOIL Method.

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Step 12.5.16.5.1

Apply the distributive property.

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{\frac{(x (x - 2) + 2 (x - 2)) \sqrt{(x + 2) (x - 2)}}{2^{3}}}{3} + C$

Step 12.5.16.5.2

Apply the distributive property.

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{\frac{(x \cdot x + x \cdot - 2 + 2 (x - 2)) \sqrt{(x + 2) (x - 2)}}{2^{3}}}{3} + C$

Step 12.5.16.5.3

Apply the distributive property.

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{\frac{(x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2) \sqrt{(x + 2) (x - 2)}}{2^{3}}}{3} + C$

Step 12.5.16.6

Combine the opposite terms in $x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2$ .

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Step 12.5.16.6.1

Reorder the factors in the terms $x \cdot - 2$ and $2 x$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{\frac{(x \cdot x - 2 x + 2 x + 2 \cdot - 2) \sqrt{(x + 2) (x - 2)}}{2^{3}}}{3} + C$

Step 12.5.16.6.2

Add $- 2 x$ and $2 x$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{\frac{(x \cdot x + 0 + 2 \cdot - 2) \sqrt{(x + 2) (x - 2)}}{2^{3}}}{3} + C$

Step 12.5.16.6.3

Add $x \cdot x$ and $0$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{\frac{(x \cdot x + 2 \cdot - 2) \sqrt{(x + 2) (x - 2)}}{2^{3}}}{3} + C$

Step 12.5.16.7

Multiply $2$ by $- 2$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{\frac{(x \cdot x - 4) \sqrt{(x + 2) (x - 2)}}{2^{3}}}{3} + C$

Step 12.5.16.8

Multiply $(x \cdot x - 4) \sqrt{(x + 2) (x - 2)}$ .

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Step 12.5.16.8.1

Raise $x$ to the power of $1$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{\frac{(x^{1} x - 4) \sqrt{(x + 2) (x - 2)}}{2^{3}}}{3} + C$

Step 12.5.16.8.2

Raise $x$ to the power of $1$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{\frac{(x^{1} x^{1} - 4) \sqrt{(x + 2) (x - 2)}}{2^{3}}}{3} + C$

Step 12.5.16.8.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{\frac{(x^{1 + 1} - 4) \sqrt{(x + 2) (x - 2)}}{2^{3}}}{3} + C$

Step 12.5.16.8.4

Add $1$ and $1$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{\frac{(x^{2} - 4) \sqrt{(x + 2) (x - 2)}}{2^{3}}}{3} + C$

Step 12.5.16.9

Apply the distributive property.

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{\frac{x^{2} \sqrt{(x + 2) (x - 2)} - 4 \sqrt{(x + 2) (x - 2)}}{2^{3}}}{3} + C$

Step 12.5.16.10

Factor $\sqrt{(x + 2) (x - 2)}$ out of $x^{2} \sqrt{(x + 2) (x - 2)} - 4 \sqrt{(x + 2) (x - 2)}$ .

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Step 12.5.16.10.1

Factor $\sqrt{(x + 2) (x - 2)}$ out of $x^{2} \sqrt{(x + 2) (x - 2)}$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{\frac{\sqrt{(x + 2) (x - 2)} x^{2} - 4 \sqrt{(x + 2) (x - 2)}}{2^{3}}}{3} + C$

Step 12.5.16.10.2

Factor $\sqrt{(x + 2) (x - 2)}$ out of $- 4 \sqrt{(x + 2) (x - 2)}$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{\frac{\sqrt{(x + 2) (x - 2)} x^{2} + \sqrt{(x + 2) (x - 2)} \cdot - 4}{2^{3}}}{3} + C$

Step 12.5.16.10.3

Factor $\sqrt{(x + 2) (x - 2)}$ out of $\sqrt{(x + 2) (x - 2)} x^{2} + \sqrt{(x + 2) (x - 2)} \cdot - 4$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{\frac{\sqrt{(x + 2) (x - 2)} (x^{2} - 4)}{2^{3}}}{3} + C$

Step 12.5.16.11

Rewrite $4$ as $2^{2}$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{\frac{\sqrt{(x + 2) (x - 2)} (x^{2} - 2^{2})}{2^{3}}}{3} + C$

Step 12.5.16.12

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{\frac{\sqrt{(x + 2) (x - 2)} (x + 2) (x - 2)}{2^{3}}}{3} + C$

Step 12.5.17

Raise $2$ to the power of $3$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{\frac{\sqrt{(x + 2) (x - 2)} (x + 2) (x - 2)}{8}}{3} + C$

Step 12.6

Simplify each term.

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Step 12.6.1

Multiply the numerator by the reciprocal of the denominator.

$4 \sqrt{(x + 2) (x - 2)} + 8 (\frac{\sqrt{(x + 2) (x - 2)} (x + 2) (x - 2)}{8} \cdot \frac{1}{3}) + C$

Step 12.6.2

Multiply $\frac{\sqrt{(x + 2) (x - 2)} (x + 2) (x - 2)}{8} \cdot \frac{1}{3}$ .

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Step 12.6.2.1

Multiply $\frac{\sqrt{(x + 2) (x - 2)} (x + 2) (x - 2)}{8}$ by $\frac{1}{3}$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{\sqrt{(x + 2) (x - 2)} (x + 2) (x - 2)}{8 \cdot 3} + C$

Step 12.6.2.2

Multiply $8$ by $3$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{\sqrt{(x + 2) (x - 2)} (x + 2) (x - 2)}{24} + C$

Step 12.6.3

Cancel the common factor of $8$ .

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Step 12.6.3.1

Factor $8$ out of $24$ .

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{\sqrt{(x + 2) (x - 2)} (x + 2) (x - 2)}{8 (3)} + C$

Step 12.6.3.2

Cancel the common factor.

$4 \sqrt{(x + 2) (x - 2)} + 8 \frac{\sqrt{(x + 2) (x - 2)} (x + 2) (x - 2)}{8 \cdot 3} + C$

Step 12.6.3.3

Rewrite the expression.

$4 \sqrt{(x + 2) (x - 2)} + \frac{\sqrt{(x + 2) (x - 2)} (x + 2) (x - 2)}{3} + C$

Step 12.7

To write $4 \sqrt{(x + 2) (x - 2)}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$4 \sqrt{(x + 2) (x - 2)} \cdot \frac{3}{3} + \frac{\sqrt{(x + 2) (x - 2)} (x + 2) (x - 2)}{3} + C$

Step 12.8

Combine $4 \sqrt{(x + 2) (x - 2)}$ and $\frac{3}{3}$ .

$\frac{4 \sqrt{(x + 2) (x - 2)} \cdot 3}{3} + \frac{\sqrt{(x + 2) (x - 2)} (x + 2) (x - 2)}{3} + C$

Step 12.9

Combine the numerators over the common denominator.

$\frac{4 \sqrt{(x + 2) (x - 2)} \cdot 3 + \sqrt{(x + 2) (x - 2)} (x + 2) (x - 2)}{3} + C$

Step 12.10

Simplify the numerator.

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Step 12.10.1

Factor $\sqrt{(x + 2) (x - 2)}$ out of $4 \sqrt{(x + 2) (x - 2)} \cdot 3 + \sqrt{(x + 2) (x - 2)} (x + 2) (x - 2)$ .

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Step 12.10.1.1

Factor $\sqrt{(x + 2) (x - 2)}$ out of $4 \sqrt{(x + 2) (x - 2)} \cdot 3$ .

$\frac{\sqrt{(x + 2) (x - 2)} (4 \cdot 3) + \sqrt{(x + 2) (x - 2)} (x + 2) (x - 2)}{3} + C$

Step 12.10.1.2

Factor $\sqrt{(x + 2) (x - 2)}$ out of $\sqrt{(x + 2) (x - 2)} (x + 2) (x - 2)$ .

$\frac{\sqrt{(x + 2) (x - 2)} (4 \cdot 3) + \sqrt{(x + 2) (x - 2)} ((x + 2) (x - 2))}{3} + C$

Step 12.10.1.3

Factor $\sqrt{(x + 2) (x - 2)}$ out of $\sqrt{(x + 2) (x - 2)} (4 \cdot 3) + \sqrt{(x + 2) (x - 2)} ((x + 2) (x - 2))$ .

$\frac{\sqrt{(x + 2) (x - 2)} (4 \cdot 3 + (x + 2) (x - 2))}{3} + C$

Step 12.10.2

Multiply $4$ by $3$ .

$\frac{\sqrt{(x + 2) (x - 2)} (12 + (x + 2) (x - 2))}{3} + C$

Step 12.10.3

Expand $(x + 2) (x - 2)$ using the FOIL Method.

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Step 12.10.3.1

Apply the distributive property.

$\frac{\sqrt{(x + 2) (x - 2)} (12 + x (x - 2) + 2 (x - 2))}{3} + C$

Step 12.10.3.2

Apply the distributive property.

$\frac{\sqrt{(x + 2) (x - 2)} (12 + x \cdot x + x \cdot - 2 + 2 (x - 2))}{3} + C$

Step 12.10.3.3

Apply the distributive property.

$\frac{\sqrt{(x + 2) (x - 2)} (12 + x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2)}{3} + C$

Step 12.10.4

Combine the opposite terms in $x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2$ .

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Step 12.10.4.1

Reorder the factors in the terms $x \cdot - 2$ and $2 x$ .

$\frac{\sqrt{(x + 2) (x - 2)} (12 + x \cdot x - 2 x + 2 x + 2 \cdot - 2)}{3} + C$

Step 12.10.4.2

Add $- 2 x$ and $2 x$ .

$\frac{\sqrt{(x + 2) (x - 2)} (12 + x \cdot x + 0 + 2 \cdot - 2)}{3} + C$

Step 12.10.4.3

Add $x \cdot x$ and $0$ .

$\frac{\sqrt{(x + 2) (x - 2)} (12 + x \cdot x + 2 \cdot - 2)}{3} + C$

Step 12.10.5

Multiply $2$ by $- 2$ .

$\frac{\sqrt{(x + 2) (x - 2)} (12 + x \cdot x - 4)}{3} + C$

Step 12.10.6

Subtract $4$ from $12$ .

$\frac{\sqrt{(x + 2) (x - 2)} (x \cdot x + 8)}{3} + C$

Step 12.10.7

Multiply $x$ by $x$ .

$\frac{\sqrt{(x + 2) (x - 2)} (x^{2} + 8)}{3} + C$