Calculus Examples

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

Step 1

Let $x = \frac{3}{2} \sec (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = \frac{3 \sec (t) \tan (t)}{2} d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\frac{3 \sec (t) \tan (t)}{2}$ is positive.

$\int \frac{\sqrt{4 {(\frac{3}{2} \sec (t))}^{2} - 9}}{\frac{3}{2} \sec (t)} \cdot \frac{3 \sec (t) \tan (t)}{2} d t$

Step 2

Simplify terms.

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Step 2.1

Simplify $\sqrt{4 {(\frac{3}{2} \sec (t))}^{2} - 9}$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Combine $\frac{3}{2}$ and $\sec (t)$ .

$\int \frac{\sqrt{4 {(\frac{3 \sec (t)}{2})}^{2} - 9}}{\frac{3}{2} \sec (t)} \cdot \frac{3 \sec (t) \tan (t)}{2} d t$

Step 2.1.1.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1.1.2.1

Apply the product rule to $\frac{3 \sec (t)}{2}$ .

$\int \frac{\sqrt{4 \frac{{(3 \sec (t))}^{2}}{2^{2}} - 9}}{\frac{3}{2} \sec (t)} \cdot \frac{3 \sec (t) \tan (t)}{2} d t$

Step 2.1.1.2.2

Apply the product rule to $3 \sec (t)$ .

$\int \frac{\sqrt{4 \frac{3^{2} \sec^{2} (t)}{2^{2}} - 9}}{\frac{3}{2} \sec (t)} \cdot \frac{3 \sec (t) \tan (t)}{2} d t$

Step 2.1.1.3

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

$\int \frac{\sqrt{4 \frac{9 \sec^{2} (t)}{2^{2}} - 9}}{\frac{3}{2} \sec (t)} \cdot \frac{3 \sec (t) \tan (t)}{2} d t$

Step 2.1.1.4

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

$\int \frac{\sqrt{4 \frac{9 \sec^{2} (t)}{4} - 9}}{\frac{3}{2} \sec (t)} \cdot \frac{3 \sec (t) \tan (t)}{2} d t$

Step 2.1.1.5

Cancel the common factor of $4$ .

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Step 2.1.1.5.1

Cancel the common factor.

$\int \frac{\sqrt{4 \frac{9 \sec^{2} (t)}{4} - 9}}{\frac{3}{2} \sec (t)} \cdot \frac{3 \sec (t) \tan (t)}{2} d t$

Step 2.1.1.5.2

Rewrite the expression.

$\int \frac{\sqrt{9 \sec^{2} (t) - 9}}{\frac{3}{2} \sec (t)} \cdot \frac{3 \sec (t) \tan (t)}{2} d t$

Step 2.1.2

Factor $9$ out of $9 \sec^{2} (t)$ .

$\int \frac{\sqrt{9 (\sec^{2} (t)) - 9}}{\frac{3}{2} \sec (t)} \cdot \frac{3 \sec (t) \tan (t)}{2} d t$

Step 2.1.3

Factor $9$ out of $- 9$ .

$\int \frac{\sqrt{9 \sec^{2} (t) + 9 \cdot - 1}}{\frac{3}{2} \sec (t)} \cdot \frac{3 \sec (t) \tan (t)}{2} d t$

Step 2.1.4

Factor $9$ out of $9 \sec^{2} (t) + 9 \cdot - 1$ .

$\int \frac{\sqrt{9 (\sec^{2} (t) - 1)}}{\frac{3}{2} \sec (t)} \cdot \frac{3 \sec (t) \tan (t)}{2} d t$

Step 2.1.5

Apply pythagorean identity.

$\int \frac{\sqrt{9 \tan^{2} (t)}}{\frac{3}{2} \sec (t)} \cdot \frac{3 \sec (t) \tan (t)}{2} d t$

Step 2.1.6

Rewrite $9 \tan^{2} (t)$ as ${(3 \tan (t))}^{2}$ .

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

Step 2.1.7

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

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

Step 2.2

Simplify.

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Step 2.2.1

Combine $\frac{3}{2}$ and $\sec (t)$ .

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

Step 2.2.2

Multiply by the reciprocal of the fraction to divide by $\frac{3 \sec (t)}{2}$ .

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

Step 2.2.3

Combine $\frac{2}{3 \sec (t)}$ and $3$ .

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

Step 2.2.4

Multiply $2$ by $3$ .

$\int \frac{6}{3 \sec (t)} \tan (t) \frac{3 \sec (t) \tan (t)}{2} d t$

Step 2.2.5

Combine $\frac{6}{3 \sec (t)}$ and $\tan (t)$ .

$\int \frac{6 \tan (t)}{3 \sec (t)} \cdot \frac{3 \sec (t) \tan (t)}{2} d t$

Step 2.2.6

Cancel the common factor of $6$ and $3$ .

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Step 2.2.6.1

Factor $3$ out of $6 \tan (t)$ .

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

Step 2.2.6.2

Cancel the common factors.

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Step 2.2.6.2.1

Factor $3$ out of $3 \sec (t)$ .

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

Step 2.2.6.2.2

Cancel the common factor.

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

Step 2.2.6.2.3

Rewrite the expression.

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

Step 2.2.7

Multiply $\frac{2 \tan (t)}{\sec (t)}$ by $\frac{3 \sec (t) \tan (t)}{2}$ .

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

Step 2.2.8

Multiply $3$ by $2$ .

$\int \frac{6 \tan (t) (\sec (t) \tan (t))}{\sec (t) \cdot 2} d t$

Step 2.2.9

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

$\int \frac{6 (\tan^{1} (t) \tan (t)) \sec (t)}{\sec (t) \cdot 2} d t$

Step 2.2.10

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

$\int \frac{6 (\tan^{1} (t) \tan^{1} (t)) \sec (t)}{\sec (t) \cdot 2} d t$

Step 2.2.11

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

$\int \frac{6 {\tan (t)}^{1 + 1} \sec (t)}{\sec (t) \cdot 2} d t$

Step 2.2.12

Add $1$ and $1$ .

$\int \frac{6 \tan^{2} (t) \sec (t)}{\sec (t) \cdot 2} d t$

Step 2.2.13

Move $2$ to the left of $\sec (t)$ .

$\int \frac{6 \tan^{2} (t) \sec (t)}{2 \cdot \sec (t)} d t$

Step 2.2.14

Cancel the common factor of $6$ and $2$ .

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Step 2.2.14.1

Factor $2$ out of $6 \tan^{2} (t) \sec (t)$ .

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

Step 2.2.14.2

Cancel the common factors.

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Step 2.2.14.2.1

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

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

Step 2.2.14.2.2

Cancel the common factor.

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

Step 2.2.14.2.3

Rewrite the expression.

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

Step 2.2.15

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

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Step 2.2.15.1

Cancel the common factor.

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

Step 2.2.15.2

Divide $3 \tan^{2} (t)$ by $1$ .

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

Step 3

Since $3$ is constant with respect to $t$ , move $3$ out of the integral.

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

Step 4

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

$3 \int - 1 + \sec^{2} (t) d t$

Step 5

Split the single integral into multiple integrals.

$3 (\int - 1 d t + \int \sec^{2} (t) d t)$

Step 6

Apply the constant rule.

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

Step 7

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

$3 (- t + C + \tan (t) + C)$

Step 8

Simplify.

$3 (- t + \tan (t)) + C$

Step 9

Replace all occurrences of $t$ with $arcsec (\frac{2 x}{3})$ .

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

Step 10

Simplify.

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Step 10.1

Simplify each term.

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Step 10.1.1

Draw a triangle in the plane with vertices $(1, \sqrt{{(\frac{2 x}{3})}^{2} - 1^{2}})$ , $(1, 0)$ , and the origin. Then $arcsec (\frac{2 x}{3})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, \sqrt{{(\frac{2 x}{3})}^{2} - 1^{2}})$ . Therefore, $\tan (arcsec (\frac{2 x}{3}))$ is $\sqrt{{(\frac{2 x}{3})}^{2} - 1}$ .

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

Step 10.1.2

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

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

Step 10.1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \frac{2 x}{3}$ and $b = 1$ .

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

Step 10.1.4

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

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

Step 10.1.5

Combine the numerators over the common denominator.

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

Step 10.1.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 10.1.7

Combine $- 1$ and $\frac{3}{3}$ .

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

Step 10.1.8

Combine the numerators over the common denominator.

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

Step 10.1.9

Multiply $- 1$ by $3$ .

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

Step 10.1.10

Multiply $\frac{2 x + 3}{3}$ by $\frac{2 x - 3}{3}$ .

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

Step 10.1.11

Multiply $3$ by $3$ .

$3 (- arcsec (\frac{2 x}{3}) + \sqrt{\frac{(2 x + 3) (2 x - 3)}{9}}) + C$

Step 10.1.12

Rewrite $\frac{(2 x + 3) (2 x - 3)}{9}$ as ${(\frac{1}{3})}^{2} ((2 x + 3) (2 x - 3))$ .

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Step 10.1.12.1

Factor the perfect power $1^{2}$ out of $(2 x + 3) (2 x - 3)$ .

$3 (- arcsec (\frac{2 x}{3}) + \sqrt{\frac{1^{2} ((2 x + 3) (2 x - 3))}{9}}) + C$

Step 10.1.12.2

Factor the perfect power $3^{2}$ out of $9$ .

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

Step 10.1.12.3

Rearrange the fraction $\frac{1^{2} ((2 x + 3) (2 x - 3))}{3^{2} \cdot 1}$ .

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

Step 10.1.13

Pull terms out from under the radical.

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

Step 10.1.14

Combine $\frac{1}{3}$ and $\sqrt{(2 x + 3) (2 x - 3)}$ .

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

Step 10.2

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

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

Step 10.3

Combine $- arcsec (\frac{2 x}{3})$ and $\frac{3}{3}$ .

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

Step 10.4

Combine the numerators over the common denominator.

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

Step 10.5

Cancel the common factor of $3$ .

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Step 10.5.1

Cancel the common factor.

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

Step 10.5.2

Rewrite the expression.

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

Step 10.6

Multiply $3$ by $- 1$ .

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

Step 11

Reorder terms.

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