Calculus Examples

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

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.

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

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 $2 \sec (t)$ .

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

Step 2.1.1.2

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

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

Step 2.1.2

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

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

Step 2.1.3

Factor $4$ out of $- 4$ .

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

Step 2.1.4

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

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

Step 2.1.5

Apply pythagorean identity.

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

Step 2.1.6

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

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

Step 2.1.7

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

$\int 2 \tan (t) (2 \sec (t) \tan (t)) d t$

Step 2.2

Simplify.

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Step 2.2.1

Multiply $2$ by $2$ .

$\int 4 \tan (t) (\sec (t) \tan (t)) d t$

Step 2.2.2

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

$\int 4 (\tan^{1} (t) \tan (t)) \sec (t) d t$

Step 2.2.3

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

$\int 4 (\tan^{1} (t) \tan^{1} (t)) \sec (t) d t$

Step 2.2.4

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

$\int 4 {\tan (t)}^{1 + 1} \sec (t) d t$

Step 2.2.5

Add $1$ and $1$ .

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

Step 3

Since $4$ is constant with respect to $t$ , move $4$ out of the integral.

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

Step 4

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

$4 \int \tan^{2} (t) \sec^{1} (t) d t$

Step 5

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

$4 \int (- 1 + \sec^{2} (t)) \sec^{1} (t) d t$

Step 6

Simplify terms.

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Step 6.1

Apply the distributive property.

$4 \int - 1 \sec^{1} (t) + \sec^{2} (t) \sec^{1} (t) d t$

Step 6.2

Simplify each term.

$4 \int - \sec (t) + \sec^{3} (t) d t$

Step 7

Split the single integral into multiple integrals.

$4 (\int - \sec (t) d t + \int \sec^{3} (t) d t)$

Step 8

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

$4 (- \int \sec (t) d t + \int \sec^{3} (t) d t)$

Step 9

The integral of $\sec (t)$ with respect to $t$ is $\ln (| \sec (t) + \tan (t) |)$ .

$4 (- (\ln (| \sec (t) + \tan (t) |) + C) + \int \sec^{3} (t) d t)$

Step 10

Factor $\sec (t)$ out of $\sec^{3} (t)$ .

$4 (- (\ln (| \sec (t) + \tan (t) |) + C) + \int \sec (t) \sec^{2} (t) d t)$

Step 11

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \sec (t)$ and $d v = \sec^{2} (t)$ .

$4 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int \tan (t) (\sec (t) \tan (t)) d t)$

Step 12

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

$4 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int \tan^{1} (t) \tan (t) \sec (t) d t)$

Step 13

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

$4 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int \tan^{1} (t) \tan^{1} (t) \sec (t) d t)$

Step 14

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

$4 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int {\tan (t)}^{1 + 1} \sec (t) d t)$

Step 15

Simplify the expression.

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Step 15.1

Add $1$ and $1$ .

$4 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int \tan^{2} (t) \sec (t) d t)$

Step 15.2

Reorder $\tan^{2} (t)$ and $\sec (t)$ .

$4 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int \sec (t) \tan^{2} (t) d t)$

Step 16

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

$4 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int \sec (t) (- 1 + \sec^{2} (t)) d t)$

Step 17

Simplify by multiplying through.

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Step 17.1

Rewrite the exponentiation as a product.

$4 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int \sec (t) (- 1 + \sec (t) \sec (t)) d t)$

Step 17.2

Apply the distributive property.

$4 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int \sec (t) \cdot - 1 + \sec (t) (\sec (t) \sec (t)) d t)$

Step 17.3

Reorder $\sec (t)$ and $- 1$ .

$4 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int - 1 \cdot \sec (t) + \sec (t) (\sec (t) \sec (t)) d t)$

Step 18

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

$4 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{1} (t) \sec (t) \sec (t) d t)$

Step 19

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

$4 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{1} (t) \sec^{1} (t) \sec (t) d t)$

Step 20

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

$4 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int - 1 \sec (t) + {\sec (t)}^{1 + 1} \sec (t) d t)$

Step 21

Add $1$ and $1$ .

$4 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{2} (t) \sec (t) d t)$

Step 22

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

$4 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{2} (t) \sec^{1} (t) d t)$

Step 23

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

$4 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int - 1 \sec (t) + {\sec (t)}^{2 + 1} d t)$

Step 24

Add $2$ and $1$ .

$4 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{3} (t) d t)$

Step 25

Split the single integral into multiple integrals.

$4 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - (\int - 1 \sec (t) d t + \int \sec^{3} (t) d t))$

Step 26

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

$4 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - (- \int \sec (t) d t + \int \sec^{3} (t) d t))$

Step 27

The integral of $\sec (t)$ with respect to $t$ is $\ln (| \sec (t) + \tan (t) |)$ .

$4 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - (- (\ln (| \sec (t) + \tan (t) |) + C) + \int \sec^{3} (t) d t))$

Step 28

Simplify by multiplying through.

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Step 28.1

Apply the distributive property.

$4 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - - (\ln (| \sec (t) + \tan (t) |) + C) - \int \sec^{3} (t) d t)$

Step 28.2

Multiply $- 1$ by $- 1$ .

$4 (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) + 1 (\ln (| \sec (t) + \tan (t) |) + C) - \int \sec^{3} (t) d t)$

Step 29

Solving for $\int \sec^{3} (t) d t$ , we find that $\int \sec^{3} (t) d t$ = $\frac{\sec (t) \tan (t) + 1 (\ln (| \sec (t) + \tan (t) |) + C)}{2}$ .

$4 (- (\ln (| \sec (t) + \tan (t) |) + C) + \frac{\sec (t) \tan (t) + 1 (\ln (| \sec (t) + \tan (t) |) + C)}{2} + C)$

Step 30

Multiply $\ln (| \sec (t) + \tan (t) |) + C$ by $1$ .

$4 (- (\ln (| \sec (t) + \tan (t) |) + C) + \frac{\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |) + C}{2} + C)$

Step 31

Simplify.

$4 \frac{- \ln (| \sec (t) + \tan (t) |) \cdot 2 + \sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)}{2} + C$

Step 32

Simplify.

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Step 32.1

Multiply $2$ by $- 1$ .

$4 \frac{- 2 \ln (| \sec (t) + \tan (t) |) + \sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)}{2} + C$

Step 32.2

Add $- 2 \ln (| \sec (t) + \tan (t) |)$ and $\ln (| \sec (t) + \tan (t) |)$ .

$4 \frac{\sec (t) \tan (t) - \ln (| \sec (t) + \tan (t) |)}{2} + C$

Step 32.3

Combine $4$ and $\frac{\sec (t) \tan (t) - \ln (| \sec (t) + \tan (t) |)}{2}$ .

$\frac{4 (\sec (t) \tan (t) - \ln (| \sec (t) + \tan (t) |))}{2} + C$

Step 32.4

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

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Step 32.4.1

Factor $2$ out of $4 (\sec (t) \tan (t) - \ln (| \sec (t) + \tan (t) |))$ .

$\frac{2 (2 (\sec (t) \tan (t) - \ln (| \sec (t) + \tan (t) |)))}{2} + C$

Step 32.4.2

Cancel the common factors.

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Step 32.4.2.1

Factor $2$ out of $2$ .

$\frac{2 (2 (\sec (t) \tan (t) - \ln (| \sec (t) + \tan (t) |)))}{2 (1)} + C$

Step 32.4.2.2

Cancel the common factor.

$\frac{2 (2 (\sec (t) \tan (t) - \ln (| \sec (t) + \tan (t) |)))}{2 \cdot 1} + C$

Step 32.4.2.3

Rewrite the expression.

$\frac{2 (\sec (t) \tan (t) - \ln (| \sec (t) + \tan (t) |))}{1} + C$

Step 32.4.2.4

Divide $2 (\sec (t) \tan (t) - \ln (| \sec (t) + \tan (t) |))$ by $1$ .

$2 (\sec (t) \tan (t) - \ln (| \sec (t) + \tan (t) |)) + C$

Step 33

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

$2 (\sec (arcsec (\frac{x}{2})) \tan (arcsec (\frac{x}{2})) - \ln (| \sec (arcsec (\frac{x}{2})) + \tan (arcsec (\frac{x}{2})) |)) + C$

Step 34

Simplify.

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Step 34.1

Simplify each term.

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Step 34.1.1

The functions secant and arcsecant are inverses.

$2 (\frac{x}{2} \tan (arcsec (\frac{x}{2})) - \ln (| \sec (arcsec (\frac{x}{2})) + \tan (arcsec (\frac{x}{2})) |)) + C$

Step 34.1.2

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 $\sqrt{{(\frac{x}{2})}^{2} - 1}$ .

$2 (\frac{x}{2} \sqrt{{(\frac{x}{2})}^{2} - 1} - \ln (| \sec (arcsec (\frac{x}{2})) + \tan (arcsec (\frac{x}{2})) |)) + C$

Step 34.1.3

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

$2 (\frac{x}{2} \sqrt{{(\frac{x}{2})}^{2} - 1^{2}} - \ln (| \sec (arcsec (\frac{x}{2})) + \tan (arcsec (\frac{x}{2})) |)) + C$

Step 34.1.4

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 $b = 1$ .

$2 (\frac{x}{2} \sqrt{(\frac{x}{2} + 1) (\frac{x}{2} - 1)} - \ln (| \sec (arcsec (\frac{x}{2})) + \tan (arcsec (\frac{x}{2})) |)) + C$

Step 34.1.5

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

$2 (\frac{x}{2} \sqrt{(\frac{x}{2} + \frac{2}{2}) (\frac{x}{2} - 1)} - \ln (| \sec (arcsec (\frac{x}{2})) + \tan (arcsec (\frac{x}{2})) |)) + C$

Step 34.1.6

Combine the numerators over the common denominator.

$2 (\frac{x}{2} \sqrt{\frac{x + 2}{2} (\frac{x}{2} - 1)} - \ln (| \sec (arcsec (\frac{x}{2})) + \tan (arcsec (\frac{x}{2})) |)) + C$

Step 34.1.7

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

$2 (\frac{x}{2} \sqrt{\frac{x + 2}{2} (\frac{x}{2} - 1 \cdot \frac{2}{2})} - \ln (| \sec (arcsec (\frac{x}{2})) + \tan (arcsec (\frac{x}{2})) |)) + C$

Step 34.1.8

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

$2 (\frac{x}{2} \sqrt{\frac{x + 2}{2} (\frac{x}{2} + \frac{- 1 \cdot 2}{2})} - \ln (| \sec (arcsec (\frac{x}{2})) + \tan (arcsec (\frac{x}{2})) |)) + C$

Step 34.1.9

Combine the numerators over the common denominator.

$2 (\frac{x}{2} \sqrt{\frac{x + 2}{2} \cdot \frac{x - 1 \cdot 2}{2}} - \ln (| \sec (arcsec (\frac{x}{2})) + \tan (arcsec (\frac{x}{2})) |)) + C$

Step 34.1.10

Multiply $- 1$ by $2$ .

$2 (\frac{x}{2} \sqrt{\frac{x + 2}{2} \cdot \frac{x - 2}{2}} - \ln (| \sec (arcsec (\frac{x}{2})) + \tan (arcsec (\frac{x}{2})) |)) + C$

Step 34.1.11

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

$2 (\frac{x}{2} \sqrt{\frac{(x + 2) (x - 2)}{2 \cdot 2}} - \ln (| \sec (arcsec (\frac{x}{2})) + \tan (arcsec (\frac{x}{2})) |)) + C$

Step 34.1.12

Multiply $2$ by $2$ .

$2 (\frac{x}{2} \sqrt{\frac{(x + 2) (x - 2)}{4}} - \ln (| \sec (arcsec (\frac{x}{2})) + \tan (arcsec (\frac{x}{2})) |)) + C$

Step 34.1.13

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

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Step 34.1.13.1

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

$2 (\frac{x}{2} \sqrt{\frac{1^{2} ((x + 2) (x - 2))}{4}} - \ln (| \sec (arcsec (\frac{x}{2})) + \tan (arcsec (\frac{x}{2})) |)) + C$

Step 34.1.13.2

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

$2 (\frac{x}{2} \sqrt{\frac{1^{2} ((x + 2) (x - 2))}{2^{2} \cdot 1}} - \ln (| \sec (arcsec (\frac{x}{2})) + \tan (arcsec (\frac{x}{2})) |)) + C$

Step 34.1.13.3

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

$2 (\frac{x}{2} \sqrt{{(\frac{1}{2})}^{2} ((x + 2) (x - 2))} - \ln (| \sec (arcsec (\frac{x}{2})) + \tan (arcsec (\frac{x}{2})) |)) + C$

Step 34.1.14

Pull terms out from under the radical.

$2 (\frac{x}{2} (\frac{1}{2} \sqrt{(x + 2) (x - 2)}) - \ln (| \sec (arcsec (\frac{x}{2})) + \tan (arcsec (\frac{x}{2})) |)) + C$

Step 34.1.15

Rewrite using the commutative property of multiplication.

$2 (\frac{1}{2} \cdot \frac{x}{2} \sqrt{(x + 2) (x - 2)} - \ln (| \sec (arcsec (\frac{x}{2})) + \tan (arcsec (\frac{x}{2})) |)) + C$

Step 34.1.16

Multiply $\frac{1}{2} \cdot \frac{x}{2}$ .

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Step 34.1.16.1

Multiply $\frac{1}{2}$ by $\frac{x}{2}$ .

$2 (\frac{x}{2 \cdot 2} \sqrt{(x + 2) (x - 2)} - \ln (| \sec (arcsec (\frac{x}{2})) + \tan (arcsec (\frac{x}{2})) |)) + C$

Step 34.1.16.2

Multiply $2$ by $2$ .

$2 (\frac{x}{4} \sqrt{(x + 2) (x - 2)} - \ln (| \sec (arcsec (\frac{x}{2})) + \tan (arcsec (\frac{x}{2})) |)) + C$

Step 34.1.17

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

$2 (\frac{x \sqrt{(x + 2) (x - 2)}}{4} - \ln (| \sec (arcsec (\frac{x}{2})) + \tan (arcsec (\frac{x}{2})) |)) + C$

Step 34.1.18

Simplify each term.

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Step 34.1.18.1

The functions secant and arcsecant are inverses.

$2 (\frac{x \sqrt{(x + 2) (x - 2)}}{4} - \ln (| \frac{x}{2} + \tan (arcsec (\frac{x}{2})) |)) + C$

Step 34.1.18.2

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

Step 34.1.18.3

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

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

Step 34.1.18.4

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 $b = 1$ .

$2 (\frac{x \sqrt{(x + 2) (x - 2)}}{4} - \ln (| \frac{x}{2} + \sqrt{(\frac{x}{2} + 1) (\frac{x}{2} - 1)} |)) + C$

Step 34.1.18.5

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

$2 (\frac{x \sqrt{(x + 2) (x - 2)}}{4} - \ln (| \frac{x}{2} + \sqrt{(\frac{x}{2} + \frac{2}{2}) (\frac{x}{2} - 1)} |)) + C$

Step 34.1.18.6

Combine the numerators over the common denominator.

$2 (\frac{x \sqrt{(x + 2) (x - 2)}}{4} - \ln (| \frac{x}{2} + \sqrt{\frac{x + 2}{2} (\frac{x}{2} - 1)} |)) + C$

Step 34.1.18.7

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

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

Step 34.1.18.8

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

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

Step 34.1.18.9

Combine the numerators over the common denominator.

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

Step 34.1.18.10

Multiply $- 1$ by $2$ .

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

Step 34.1.18.11

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

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

Step 34.1.18.12

Multiply $2$ by $2$ .

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

Step 34.1.18.13

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

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Step 34.1.18.13.1

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

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

Step 34.1.18.13.2

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

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

Step 34.1.18.13.3

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

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

Step 34.1.18.14

Pull terms out from under the radical.

$2 (\frac{x \sqrt{(x + 2) (x - 2)}}{4} - \ln (| \frac{x}{2} + \frac{1}{2} \sqrt{(x + 2) (x - 2)} |)) + C$

Step 34.1.18.15

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

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

Step 34.1.19

Combine the numerators over the common denominator.

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

Step 34.1.20

Remove non-negative terms from the absolute value.

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

Step 34.2

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

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

Step 34.3

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

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

Step 34.4

Combine the numerators over the common denominator.

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

Step 34.5

Multiply $4$ by $- 1$ .

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

Step 34.6

Cancel the common factor of $2$ .

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Step 34.6.1

Factor $2$ out of $4$ .

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

Step 34.6.2

Cancel the common factor.

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

Step 34.6.3

Rewrite the expression.

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

Step 35

Reorder terms.

$\frac{1}{2} (x \sqrt{(x + 2) (x - 2)} - 4 \ln (\frac{1}{2} | x + \sqrt{(x + 2) (x - 2)} |)) + C$