Calculus Examples

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

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.

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

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 $2 \sec (t)$ .

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

Step 2.1.1.2

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

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

Step 2.1.2

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

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

Step 2.1.3

Factor $4$ out of $- 4$ .

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

Step 2.1.4

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

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

Step 2.1.5

Apply pythagorean identity.

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

Step 2.1.6

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

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

Step 2.1.7

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Cancel the common factor.

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

Step 2.2.3

Rewrite the expression.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Add $1$ and $1$ .

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

Step 7

Since $2$ is constant with respect to $t$ , move $2$ out of the integral.

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

Step 8

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

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

Step 9

Split the single integral into multiple integrals.

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

Step 10

Apply the constant rule.

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

Step 11

Since the derivative of $\tan (t)$ is $\sec^{2} (t)$ , the integral of $\sec^{2} (t)$ is $\tan (t)$ .

$2 (- t + C + \tan (t) + C)$

Step 12

Simplify.

$2 (- t + \tan (t)) + C$

Step 13

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

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

Step 14

Simplify.

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Step 14.1

Simplify each term.

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Step 14.1.1

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

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

Step 14.1.2

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

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

Step 14.1.3

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 $b = 1$ .

$2 (- arcsec (\frac{x}{2}) + \sqrt{(\frac{x}{2} + 1) (\frac{x}{2} - 1)}) + C$

Step 14.1.4

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

$2 (- arcsec (\frac{x}{2}) + \sqrt{(\frac{x}{2} + \frac{2}{2}) (\frac{x}{2} - 1)}) + C$

Step 14.1.5

Combine the numerators over the common denominator.

$2 (- arcsec (\frac{x}{2}) + \sqrt{\frac{x + 2}{2} (\frac{x}{2} - 1)}) + C$

Step 14.1.6

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

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

Step 14.1.7

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

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

Step 14.1.8

Combine the numerators over the common denominator.

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

Step 14.1.9

Multiply $- 1$ by $2$ .

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

Step 14.1.10

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

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

Step 14.1.11

Multiply $2$ by $2$ .

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

Step 14.1.12

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

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Step 14.1.12.1

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

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

Step 14.1.12.2

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

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

Step 14.1.12.3

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

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

Step 14.1.13

Pull terms out from under the radical.

$2 (- arcsec (\frac{x}{2}) + \frac{1}{2} \sqrt{(x + 2) (x - 2)}) + C$

Step 14.1.14

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

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

Step 14.2

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

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

Step 14.3

Combine $- arcsec (\frac{x}{2})$ and $\frac{2}{2}$ .

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

Step 14.4

Combine the numerators over the common denominator.

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

Step 14.5

Cancel the common factor of $2$ .

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Step 14.5.1

Cancel the common factor.

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

Step 14.5.2

Rewrite the expression.

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

Step 14.6

Multiply $2$ by $- 1$ .

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

Step 15

Reorder terms.

$- 2 arcsec (\frac{1}{2} x) + \sqrt{(x + 2) (x - 2)} + C$