Calculus Examples

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

Step 1

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

$\int \frac{1}{\sqrt{9 {(\frac{2}{3} \sec (t))}^{2} - 4}} \cdot \frac{2 \sec (t) \tan (t)}{3} d t$

Step 2

Simplify terms.

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Step 2.1

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

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

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

$\int \frac{1}{\sqrt{9 {(\frac{2 \sec (t)}{3})}^{2} - 4}} \cdot \frac{2 \sec (t) \tan (t)}{3} d t$

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

$\int \frac{1}{\sqrt{9 \frac{{(2 \sec (t))}^{2}}{3^{2}} - 4}} \cdot \frac{2 \sec (t) \tan (t)}{3} d t$

Step 2.1.1.2.2

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

$\int \frac{1}{\sqrt{9 \frac{2^{2} \sec^{2} (t)}{3^{2}} - 4}} \cdot \frac{2 \sec (t) \tan (t)}{3} d t$

Step 2.1.1.3

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

$\int \frac{1}{\sqrt{9 \frac{4 \sec^{2} (t)}{3^{2}} - 4}} \cdot \frac{2 \sec (t) \tan (t)}{3} d t$

Step 2.1.1.4

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

$\int \frac{1}{\sqrt{9 \frac{4 \sec^{2} (t)}{9} - 4}} \cdot \frac{2 \sec (t) \tan (t)}{3} d t$

Step 2.1.1.5

Cancel the common factor of $9$ .

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Step 2.1.1.5.1

Cancel the common factor.

$\int \frac{1}{\sqrt{9 \frac{4 \sec^{2} (t)}{9} - 4}} \cdot \frac{2 \sec (t) \tan (t)}{3} d t$

Step 2.1.1.5.2

Rewrite the expression.

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

Step 2.1.2

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

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

Step 2.1.3

Factor $4$ out of $- 4$ .

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

Step 2.1.4

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

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

Step 2.1.5

Apply pythagorean identity.

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

Step 2.1.6

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

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

Step 2.1.7

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

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

Step 2.2

Simplify.

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Step 2.2.1

Multiply $\frac{1}{2 \tan (t)}$ by $\frac{2 \sec (t) \tan (t)}{3}$ .

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

Step 2.2.2

Multiply $3$ by $2$ .

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

Step 2.2.3

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

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Step 2.2.3.1

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

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

Step 2.2.3.2

Cancel the common factors.

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Step 2.2.3.2.1

Factor $2$ out of $6 \tan (t)$ .

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

Step 2.2.3.2.2

Cancel the common factor.

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

Step 2.2.3.2.3

Rewrite the expression.

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

Step 2.2.4

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

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Step 2.2.4.1

Cancel the common factor.

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

Step 2.2.4.2

Rewrite the expression.

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

Step 3

Since $\frac{1}{3}$ is constant with respect to $t$ , move $\frac{1}{3}$ out of the integral.

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

Step 4

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

$\frac{1}{3} (\ln (| \sec (t) + \tan (t) |) + C)$

Step 5

Simplify.

$\frac{1}{3} \ln (| \sec (t) + \tan (t) |) + C$

Step 6

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

$\frac{1}{3} \ln (| \sec (arcsec (\frac{3 x}{2})) + \tan (arcsec (\frac{3 x}{2})) |) + C$

Step 7

Reorder terms.

$\frac{1}{3} \ln (| \sec (arcsec (\frac{3}{2} x)) + \tan (arcsec (\frac{3}{2} x)) |) + C$