Calculus Examples

$\int \frac{1}{x \sqrt{9 x^{2} - 7}} d x$

Step 1

Let $x = \frac{\sqrt{7}}{3} \sec (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\frac{\sqrt{7} \sec (t) \tan (t)}{3}$ is positive.

$\int \frac{1}{\frac{\sqrt{7}}{3} \sec (t) \sqrt{9 {(\frac{\sqrt{7}}{3} \sec (t))}^{2} - 7}} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

Step 2

Simplify terms.

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Step 2.1

Simplify $\sqrt{9 {(\frac{\sqrt{7}}{3} \sec (t))}^{2} - 7}$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Combine $\frac{\sqrt{7}}{3}$ and $\sec (t)$ .

$\int \frac{1}{\frac{\sqrt{7}}{3} \sec (t) \sqrt{9 {(\frac{\sqrt{7} \sec (t)}{3})}^{2} - 7}} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

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 $\frac{\sqrt{7} \sec (t)}{3}$ .

$\int \frac{1}{\frac{\sqrt{7}}{3} \sec (t) \sqrt{9 \frac{{(\sqrt{7} \sec (t))}^{2}}{3^{2}} - 7}} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

Step 2.1.1.2.2

Apply the product rule to $\sqrt{7} \sec (t)$ .

$\int \frac{1}{\frac{\sqrt{7}}{3} \sec (t) \sqrt{9 \frac{{\sqrt{7}}^{2} \sec^{2} (t)}{3^{2}} - 7}} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

Step 2.1.1.3

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 2.1.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\int \frac{1}{\frac{\sqrt{7}}{3} \sec (t) \sqrt{9 \frac{{(7^{\frac{1}{2}})}^{2} \sec^{2} (t)}{3^{2}} - 7}} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

Step 2.1.1.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int \frac{1}{\frac{\sqrt{7}}{3} \sec (t) \sqrt{9 \frac{7^{\frac{1}{2} \cdot 2} \sec^{2} (t)}{3^{2}} - 7}} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

Step 2.1.1.3.3

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

$\int \frac{1}{\frac{\sqrt{7}}{3} \sec (t) \sqrt{9 \frac{7^{\frac{2}{2}} \sec^{2} (t)}{3^{2}} - 7}} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

Step 2.1.1.3.4

Cancel the common factor of $2$ .

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Step 2.1.1.3.4.1

Cancel the common factor.

$\int \frac{1}{\frac{\sqrt{7}}{3} \sec (t) \sqrt{9 \frac{7^{\frac{2}{2}} \sec^{2} (t)}{3^{2}} - 7}} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

Step 2.1.1.3.4.2

Rewrite the expression.

$\int \frac{1}{\frac{\sqrt{7}}{3} \sec (t) \sqrt{9 \frac{7^{1} \sec^{2} (t)}{3^{2}} - 7}} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

Step 2.1.1.3.5

Evaluate the exponent.

$\int \frac{1}{\frac{\sqrt{7}}{3} \sec (t) \sqrt{9 \frac{7 \sec^{2} (t)}{3^{2}} - 7}} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

Step 2.1.1.4

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

$\int \frac{1}{\frac{\sqrt{7}}{3} \sec (t) \sqrt{9 \frac{7 \sec^{2} (t)}{9} - 7}} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

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.

$\int \frac{1}{\frac{\sqrt{7}}{3} \sec (t) \sqrt{9 \frac{7 \sec^{2} (t)}{9} - 7}} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

Step 2.1.1.5.2

Rewrite the expression.

$\int \frac{1}{\frac{\sqrt{7}}{3} \sec (t) \sqrt{7 \sec^{2} (t) - 7}} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

Step 2.1.2

Factor $7$ out of $7 \sec^{2} (t)$ .

$\int \frac{1}{\frac{\sqrt{7}}{3} \sec (t) \sqrt{7 (\sec^{2} (t)) - 7}} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

Step 2.1.3

Factor $7$ out of $- 7$ .

$\int \frac{1}{\frac{\sqrt{7}}{3} \sec (t) \sqrt{7 \sec^{2} (t) + 7 \cdot - 1}} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

Step 2.1.4

Factor $7$ out of $7 \sec^{2} (t) + 7 \cdot - 1$ .

$\int \frac{1}{\frac{\sqrt{7}}{3} \sec (t) \sqrt{7 (\sec^{2} (t) - 1)}} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

Step 2.1.5

Apply pythagorean identity.

$\int \frac{1}{\frac{\sqrt{7}}{3} \sec (t) \sqrt{7 \tan^{2} (t)}} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

Step 2.1.6

Reorder $7$ and $\tan^{2} (t)$ .

$\int \frac{1}{\frac{\sqrt{7}}{3} \sec (t) \sqrt{\tan^{2} (t) \cdot 7}} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

Step 2.1.7

Pull terms out from under the radical.

$\int \frac{1}{\frac{\sqrt{7}}{3} \sec (t) (\tan (t) \sqrt{7})} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

Step 2.2

Simplify.

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Step 2.2.1

Combine $\frac{\sqrt{7}}{3}$ and $\sec (t)$ .

$\int \frac{1}{\frac{\sqrt{7} \sec (t)}{3} (\tan (t) \sqrt{7})} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

Step 2.2.2

Combine $\tan (t)$ and $\frac{\sqrt{7} \sec (t)}{3}$ .

$\int \frac{1}{\frac{\tan (t) (\sqrt{7} \sec (t))}{3} \sqrt{7}} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

Step 2.2.3

Combine $\frac{\tan (t) (\sqrt{7} \sec (t))}{3}$ and $\sqrt{7}$ .

$\int \frac{1}{\frac{\tan (t) (\sqrt{7} \sec (t)) \sqrt{7}}{3}} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

Step 2.2.4

Raise $\sqrt{7}$ to the power of $1$ .

$\int \frac{1}{\frac{\tan (t) ({\sqrt{7}}^{1} \sqrt{7} \sec (t))}{3}} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

Step 2.2.5

Raise $\sqrt{7}$ to the power of $1$ .

$\int \frac{1}{\frac{\tan (t) ({\sqrt{7}}^{1} {\sqrt{7}}^{1} \sec (t))}{3}} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

Step 2.2.6

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

$\int \frac{1}{\frac{\tan (t) ({\sqrt{7}}^{1 + 1} \sec (t))}{3}} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

Step 2.2.7

Add $1$ and $1$ .

$\int \frac{1}{\frac{\tan (t) ({\sqrt{7}}^{2} \sec (t))}{3}} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

Step 2.2.8

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 2.2.8.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\int \frac{1}{\frac{\tan (t) ({(7^{\frac{1}{2}})}^{2} \sec (t))}{3}} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

Step 2.2.8.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int \frac{1}{\frac{\tan (t) (7^{\frac{1}{2} \cdot 2} \sec (t))}{3}} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

Step 2.2.8.3

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

$\int \frac{1}{\frac{\tan (t) (7^{\frac{2}{2}} \sec (t))}{3}} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

Step 2.2.8.4

Cancel the common factor of $2$ .

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Step 2.2.8.4.1

Cancel the common factor.

$\int \frac{1}{\frac{\tan (t) (7^{\frac{2}{2}} \sec (t))}{3}} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

Step 2.2.8.4.2

Rewrite the expression.

$\int \frac{1}{\frac{\tan (t) (7^{1} \sec (t))}{3}} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

Step 2.2.8.5

Evaluate the exponent.

$\int \frac{1}{\frac{\tan (t) (7 \sec (t))}{3}} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

Step 2.2.9

Move $7$ to the left of $\tan (t)$ .

$\int \frac{1}{\frac{7 \cdot \tan (t) \sec (t)}{3}} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

Step 2.2.10

Multiply by the reciprocal of the fraction to divide by $\frac{7 \tan (t) \sec (t)}{3}$ .

$\int 1 \frac{3}{7 \tan (t) \sec (t)} \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

Step 2.2.11

Multiply $\frac{3}{7 \tan (t) \sec (t)}$ by $1$ .

$\int \frac{3}{7 \tan (t) \sec (t)} \cdot \frac{\sqrt{7} \sec (t) \tan (t)}{3} d t$

Step 2.2.12

Multiply $\frac{3}{7 \tan (t) \sec (t)}$ by $\frac{\sqrt{7} \sec (t) \tan (t)}{3}$ .

$\int \frac{3 (\sqrt{7} \sec (t) \tan (t))}{7 \tan (t) \sec (t) \cdot 3} d t$

Step 2.2.13

Multiply $3$ by $7$ .

$\int \frac{3 (\sqrt{7} \sec (t) \tan (t))}{21 \tan (t) \sec (t)} d t$

Step 2.2.14

Cancel the common factors.

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Step 2.2.14.1

Factor $3$ out of $21 \tan (t) \sec (t)$ .

$\int \frac{3 (\sqrt{7} \sec (t) \tan (t))}{3 (7 \tan (t) \sec (t))} d t$

Step 2.2.14.2

Cancel the common factor.

$\int \frac{3 (\sqrt{7} \sec (t) \tan (t))}{3 (7 \tan (t) \sec (t))} d t$

Step 2.2.14.3

Rewrite the expression.

$\int \frac{\sqrt{7} \sec (t) \tan (t)}{7 \tan (t) \sec (t)} d t$

Step 2.2.15

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

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Step 2.2.15.1

Cancel the common factor.

$\int \frac{\sqrt{7} \sec (t) \tan (t)}{7 \tan (t) \sec (t)} d t$

Step 2.2.15.2

Rewrite the expression.

$\int \frac{\sqrt{7} \tan (t)}{7 \tan (t)} d t$

Step 2.2.16

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

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Step 2.2.16.1

Cancel the common factor.

$\int \frac{\sqrt{7} \tan (t)}{7 \tan (t)} d t$

Step 2.2.16.2

Rewrite the expression.

$\int \frac{\sqrt{7}}{7} d t$

Step 3

Apply the constant rule.

$\frac{\sqrt{7}}{7} t + C$

Step 4

Simplify the answer.

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Step 4.1

Replace all occurrences of $t$ with $arcsec (\frac{3 x}{\sqrt{7}})$ .

$\frac{\sqrt{7}}{7} arcsec (\frac{3 x}{\sqrt{7}}) + C$

Step 4.2

Reorder terms.

$\frac{\sqrt{7}}{7} arcsec (\frac{3}{\sqrt{7}} x) + C$