Calculus Examples

$\frac{1}{27} {(9 x^{2} + 6)}^{\frac{3}{2}}$

Step 1

Write $\frac{1}{27} {(9 x^{2} + 6)}^{\frac{3}{2}}$ as a function.

$f (x) = \frac{1}{27} {(9 x^{2} + 6)}^{\frac{3}{2}}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int \frac{1}{27} \cdot {(9 x^{2} + 6)}^{\frac{3}{2}} d x$

Step 4

Since $\frac{1}{27}$ is constant with respect to $x$ , move $\frac{1}{27}$ out of the integral.

$\frac{1}{27} \int {(9 x^{2} + 6)}^{\frac{3}{2}} d x$

Step 5

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\frac{1}{27} \int \sqrt{{(9 x^{2} + 6)}^{3}} d x$

Step 6

Let $x = \frac{\sqrt{6}}{3} \tan (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = \frac{\sqrt{6} \sec^{2} (t)}{3} d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\frac{\sqrt{6} \sec^{2} (t)}{3}$ is positive.

$\frac{1}{27} \int \sqrt{{(9 {(\frac{\sqrt{6}}{3} \tan (t))}^{2} + {\sqrt{6}}^{2})}^{3}} \frac{\sqrt{6} \sec^{2} (t)}{3} d t$

Step 7

Simplify terms.

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Step 7.1

Simplify $\sqrt{{(9 {(\frac{\sqrt{6}}{3} \tan (t))}^{2} + {\sqrt{6}}^{2})}^{3}}$ .

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Step 7.1.1

Simplify each term.

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Step 7.1.1.1

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

$\frac{1}{27} \int \sqrt{{(9 {(\frac{\sqrt{6} \tan (t)}{3})}^{2} + {\sqrt{6}}^{2})}^{3}} \frac{\sqrt{6} \sec^{2} (t)}{3} d t$

Step 7.1.1.2

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

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Step 7.1.1.2.1

Apply the product rule to $\frac{\sqrt{6} \tan (t)}{3}$ .

$\frac{1}{27} \int \sqrt{{(9 \frac{{(\sqrt{6} \tan (t))}^{2}}{3^{2}} + {\sqrt{6}}^{2})}^{3}} \frac{\sqrt{6} \sec^{2} (t)}{3} d t$

Step 7.1.1.2.2

Apply the product rule to $\sqrt{6} \tan (t)$ .

$\frac{1}{27} \int \sqrt{{(9 \frac{{\sqrt{6}}^{2} \tan^{2} (t)}{3^{2}} + {\sqrt{6}}^{2})}^{3}} \frac{\sqrt{6} \sec^{2} (t)}{3} d t$

Step 7.1.1.3

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 7.1.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{1}{27} \int \sqrt{{(9 \frac{{(6^{\frac{1}{2}})}^{2} \tan^{2} (t)}{3^{2}} + {\sqrt{6}}^{2})}^{3}} \frac{\sqrt{6} \sec^{2} (t)}{3} d t$

Step 7.1.1.3.2

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

$\frac{1}{27} \int \sqrt{{(9 \frac{6^{\frac{1}{2} \cdot 2} \tan^{2} (t)}{3^{2}} + {\sqrt{6}}^{2})}^{3}} \frac{\sqrt{6} \sec^{2} (t)}{3} d t$

Step 7.1.1.3.3

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

$\frac{1}{27} \int \sqrt{{(9 \frac{6^{\frac{2}{2}} \tan^{2} (t)}{3^{2}} + {\sqrt{6}}^{2})}^{3}} \frac{\sqrt{6} \sec^{2} (t)}{3} d t$

Step 7.1.1.3.4

Cancel the common factor of $2$ .

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Step 7.1.1.3.4.1

Cancel the common factor.

$\frac{1}{27} \int \sqrt{{(9 \frac{6^{\frac{2}{2}} \tan^{2} (t)}{3^{2}} + {\sqrt{6}}^{2})}^{3}} \frac{\sqrt{6} \sec^{2} (t)}{3} d t$

Step 7.1.1.3.4.2

Rewrite the expression.

$\frac{1}{27} \int \sqrt{{(9 \frac{6^{1} \tan^{2} (t)}{3^{2}} + {\sqrt{6}}^{2})}^{3}} \frac{\sqrt{6} \sec^{2} (t)}{3} d t$

Step 7.1.1.3.5

Evaluate the exponent.

$\frac{1}{27} \int \sqrt{{(9 \frac{6 \tan^{2} (t)}{3^{2}} + {\sqrt{6}}^{2})}^{3}} \frac{\sqrt{6} \sec^{2} (t)}{3} d t$

Step 7.1.1.4

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

$\frac{1}{27} \int \sqrt{{(9 \frac{6 \tan^{2} (t)}{9} + {\sqrt{6}}^{2})}^{3}} \frac{\sqrt{6} \sec^{2} (t)}{3} d t$

Step 7.1.1.5

Cancel the common factor of $9$ .

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Step 7.1.1.5.1

Cancel the common factor.

$\frac{1}{27} \int \sqrt{{(9 \frac{6 \tan^{2} (t)}{9} + {\sqrt{6}}^{2})}^{3}} \frac{\sqrt{6} \sec^{2} (t)}{3} d t$

Step 7.1.1.5.2

Rewrite the expression.

$\frac{1}{27} \int \sqrt{{(6 \tan^{2} (t) + {\sqrt{6}}^{2})}^{3}} \frac{\sqrt{6} \sec^{2} (t)}{3} d t$

Step 7.1.1.6

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 7.1.1.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{1}{27} \int \sqrt{{(6 \tan^{2} (t) + {(6^{\frac{1}{2}})}^{2})}^{3}} \frac{\sqrt{6} \sec^{2} (t)}{3} d t$

Step 7.1.1.6.2

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

$\frac{1}{27} \int \sqrt{{(6 \tan^{2} (t) + 6^{\frac{1}{2} \cdot 2})}^{3}} \frac{\sqrt{6} \sec^{2} (t)}{3} d t$

Step 7.1.1.6.3

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

$\frac{1}{27} \int \sqrt{{(6 \tan^{2} (t) + 6^{\frac{2}{2}})}^{3}} \frac{\sqrt{6} \sec^{2} (t)}{3} d t$

Step 7.1.1.6.4

Cancel the common factor of $2$ .

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Step 7.1.1.6.4.1

Cancel the common factor.

$\frac{1}{27} \int \sqrt{{(6 \tan^{2} (t) + 6^{\frac{2}{2}})}^{3}} \frac{\sqrt{6} \sec^{2} (t)}{3} d t$

Step 7.1.1.6.4.2

Rewrite the expression.

$\frac{1}{27} \int \sqrt{{(6 \tan^{2} (t) + 6^{1})}^{3}} \frac{\sqrt{6} \sec^{2} (t)}{3} d t$

Step 7.1.1.6.5

Evaluate the exponent.

$\frac{1}{27} \int \sqrt{{(6 \tan^{2} (t) + 6)}^{3}} \frac{\sqrt{6} \sec^{2} (t)}{3} d t$

Step 7.1.2

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

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Step 7.1.2.1

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

$\frac{1}{27} \int \sqrt{{(6 (\tan^{2} (t)) + 6)}^{3}} \frac{\sqrt{6} \sec^{2} (t)}{3} d t$

Step 7.1.2.2

Factor $6$ out of $6$ .

$\frac{1}{27} \int \sqrt{{(6 (\tan^{2} (t)) + 6 (1))}^{3}} \frac{\sqrt{6} \sec^{2} (t)}{3} d t$

Step 7.1.2.3

Factor $6$ out of $6 (\tan^{2} (t)) + 6 (1)$ .

$\frac{1}{27} \int \sqrt{{(6 (\tan^{2} (t) + 1))}^{3}} \frac{\sqrt{6} \sec^{2} (t)}{3} d t$

Step 7.1.3

Apply pythagorean identity.

$\frac{1}{27} \int \sqrt{{(6 \sec^{2} (t))}^{3}} \frac{\sqrt{6} \sec^{2} (t)}{3} d t$

Step 7.1.4

Apply the product rule to $6 \sec^{2} (t)$ .

$\frac{1}{27} \int \sqrt{6^{3} {(\sec^{2} (t))}^{3}} \frac{\sqrt{6} \sec^{2} (t)}{3} d t$

Step 7.1.5

Raise $6$ to the power of $3$ .

$\frac{1}{27} \int \sqrt{216 {(\sec^{2} (t))}^{3}} \frac{\sqrt{6} \sec^{2} (t)}{3} d t$

Step 7.1.6

Multiply the exponents in ${(\sec^{2} (t))}^{3}$ .

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Step 7.1.6.1

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

$\frac{1}{27} \int \sqrt{216 {\sec (t)}^{2 \cdot 3}} \frac{\sqrt{6} \sec^{2} (t)}{3} d t$

Step 7.1.6.2

Multiply $2$ by $3$ .

$\frac{1}{27} \int \sqrt{216 \sec^{6} (t)} \frac{\sqrt{6} \sec^{2} (t)}{3} d t$

Step 7.1.7

Rewrite $216 \sec^{6} (t)$ as ${(6 \sec^{3} (t))}^{2} \cdot 6$ .

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Step 7.1.7.1

Factor $36$ out of $216$ .

$\frac{1}{27} \int \sqrt{36 (6) \sec^{6} (t)} \frac{\sqrt{6} \sec^{2} (t)}{3} d t$

Step 7.1.7.2

Rewrite $36$ as $6^{2}$ .

$\frac{1}{27} \int \sqrt{6^{2} \cdot 6 \sec^{6} (t)} \frac{\sqrt{6} \sec^{2} (t)}{3} d t$

Step 7.1.7.3

Rewrite $\sec^{6} (t)$ as ${(\sec^{3} (t))}^{2}$ .

$\frac{1}{27} \int \sqrt{6^{2} \cdot 6 {(\sec^{3} (t))}^{2}} \frac{\sqrt{6} \sec^{2} (t)}{3} d t$

Step 7.1.7.4

Move $6$ .

$\frac{1}{27} \int \sqrt{6^{2} {(\sec^{3} (t))}^{2} \cdot 6} \frac{\sqrt{6} \sec^{2} (t)}{3} d t$

Step 7.1.7.5

Rewrite $6^{2} {(\sec^{3} (t))}^{2}$ as ${(6 \sec^{3} (t))}^{2}$ .

$\frac{1}{27} \int \sqrt{{(6 \sec^{3} (t))}^{2} \cdot 6} \frac{\sqrt{6} \sec^{2} (t)}{3} d t$

Step 7.1.8

Pull terms out from under the radical.

$\frac{1}{27} \int 6 \sec^{3} (t) \sqrt{6} \frac{\sqrt{6} \sec^{2} (t)}{3} d t$

Step 7.2

Simplify.

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Step 7.2.1

Combine $\frac{\sqrt{6} \sec^{2} (t)}{3}$ and $6$ .

$\frac{1}{27} \int \frac{\sqrt{6} \sec^{2} (t) \cdot 6}{3} \sec^{3} (t) \sqrt{6} d t$

Step 7.2.2

Combine $\frac{\sqrt{6} \sec^{2} (t) \cdot 6}{3}$ and $\sec^{3} (t)$ .

$\frac{1}{27} \int \frac{\sqrt{6} \sec^{2} (t) \cdot 6 \sec^{3} (t)}{3} \sqrt{6} d t$

Step 7.2.3

Multiply $\sec^{2} (t)$ by $\sec^{3} (t)$ by adding the exponents.

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Step 7.2.3.1

Move $\sec^{3} (t)$ .

$\frac{1}{27} \int \frac{\sqrt{6} (\sec^{3} (t) \sec^{2} (t)) \cdot 6}{3} \sqrt{6} d t$

Step 7.2.3.2

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

$\frac{1}{27} \int \frac{\sqrt{6} {\sec (t)}^{3 + 2} \cdot 6}{3} \sqrt{6} d t$

Step 7.2.3.3

Add $3$ and $2$ .

$\frac{1}{27} \int \frac{\sqrt{6} \sec^{5} (t) \cdot 6}{3} \sqrt{6} d t$

Step 7.2.4

Combine $\frac{\sqrt{6} \sec^{5} (t) \cdot 6}{3}$ and $\sqrt{6}$ .

$\frac{1}{27} \int \frac{\sqrt{6} \sec^{5} (t) \cdot 6 \sqrt{6}}{3} d t$

Step 7.2.5

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{1}{27} \int \frac{\sec^{5} (t) \cdot 6 ({\sqrt{6}}^{1} \sqrt{6})}{3} d t$

Step 7.2.6

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{1}{27} \int \frac{\sec^{5} (t) \cdot 6 ({\sqrt{6}}^{1} {\sqrt{6}}^{1})}{3} d t$

Step 7.2.7

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

$\frac{1}{27} \int \frac{\sec^{5} (t) \cdot 6 {\sqrt{6}}^{1 + 1}}{3} d t$

Step 7.2.8

Add $1$ and $1$ .

$\frac{1}{27} \int \frac{\sec^{5} (t) \cdot 6 {\sqrt{6}}^{2}}{3} d t$

Step 7.2.9

Move $6$ to the left of $\sec^{5} (t)$ .

$\frac{1}{27} \int \frac{6 \cdot \sec^{5} (t) {\sqrt{6}}^{2}}{3} d t$

Step 7.2.10

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 7.2.10.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{1}{27} \int \frac{6 \sec^{5} (t) {(6^{\frac{1}{2}})}^{2}}{3} d t$

Step 7.2.10.2

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

$\frac{1}{27} \int \frac{6 \sec^{5} (t) \cdot 6^{\frac{1}{2} \cdot 2}}{3} d t$

Step 7.2.10.3

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

$\frac{1}{27} \int \frac{6 \sec^{5} (t) \cdot 6^{\frac{2}{2}}}{3} d t$

Step 7.2.10.4

Cancel the common factor of $2$ .

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Step 7.2.10.4.1

Cancel the common factor.

$\frac{1}{27} \int \frac{6 \sec^{5} (t) \cdot 6^{\frac{2}{2}}}{3} d t$

Step 7.2.10.4.2

Rewrite the expression.

$\frac{1}{27} \int \frac{6 \sec^{5} (t) \cdot 6^{1}}{3} d t$

Step 7.2.10.5

Evaluate the exponent.

$\frac{1}{27} \int \frac{6 \sec^{5} (t) \cdot 6}{3} d t$

Step 7.2.11

Multiply $6$ by $6$ .

$\frac{1}{27} \int \frac{36 \sec^{5} (t)}{3} d t$

Step 7.2.12

Cancel the common factor of $36$ and $3$ .

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Step 7.2.12.1

Factor $3$ out of $36 \sec^{5} (t)$ .

$\frac{1}{27} \int \frac{3 (12 \sec^{5} (t))}{3} d t$

Step 7.2.12.2

Cancel the common factors.

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Step 7.2.12.2.1

Factor $3$ out of $3$ .

$\frac{1}{27} \int \frac{3 (12 \sec^{5} (t))}{3 (1)} d t$

Step 7.2.12.2.2

Cancel the common factor.

$\frac{1}{27} \int \frac{3 (12 \sec^{5} (t))}{3 \cdot 1} d t$

Step 7.2.12.2.3

Rewrite the expression.

$\frac{1}{27} \int \frac{12 \sec^{5} (t)}{1} d t$

Step 7.2.12.2.4

Divide $12 \sec^{5} (t)$ by $1$ .

$\frac{1}{27} \int 12 \sec^{5} (t) d t$

Step 8

Since $12$ is constant with respect to $t$ , move $12$ out of the integral.

$\frac{1}{27} (12 \int \sec^{5} (t) d t)$

Step 9

Simplify.

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Step 9.1

Combine $12$ and $\frac{1}{27}$ .

$\frac{12}{27} \int \sec^{5} (t) d t$

Step 9.2

Cancel the common factor of $12$ and $27$ .

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Step 9.2.1

Factor $3$ out of $12$ .

$\frac{3 (4)}{27} \int \sec^{5} (t) d t$

Step 9.2.2

Cancel the common factors.

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Step 9.2.2.1

Factor $3$ out of $27$ .

$\frac{3 \cdot 4}{3 \cdot 9} \int \sec^{5} (t) d t$

Step 9.2.2.2

Cancel the common factor.

$\frac{3 \cdot 4}{3 \cdot 9} \int \sec^{5} (t) d t$

Step 9.2.2.3

Rewrite the expression.

$\frac{4}{9} \int \sec^{5} (t) d t$

Step 10

Apply the reduction formula.

$\frac{4}{9} (\frac{\tan (t) \sec^{3} (t)}{4} + \frac{3}{4} \int \sec^{3} (t) d t)$

Step 11

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

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

Step 12

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

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

Step 13

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

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

Step 14

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

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

Step 15

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

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

Step 16

Simplify the expression.

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Step 16.1

Add $1$ and $1$ .

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

Step 16.2

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

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

Step 17

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

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

Step 18

Simplify by multiplying through.

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Step 18.1

Rewrite the exponentiation as a product.

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

Step 18.2

Apply the distributive property.

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

Step 18.3

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

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

Step 19

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

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

Step 20

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

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

Step 21

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

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

Step 22

Add $1$ and $1$ .

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

Step 23

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

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

Step 24

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

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

Step 25

Add $2$ and $1$ .

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

Step 26

Split the single integral into multiple integrals.

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

Step 27

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

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

Step 28

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

$\frac{4}{9} (\frac{\tan (t) \sec^{3} (t)}{4} + \frac{3}{4} (\sec (t) \tan (t) - (- (\ln (| \sec (t) + \tan (t) |) + C) + \int \sec^{3} (t) d t)))$

Step 29

Simplify by multiplying through.

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Step 29.1

Apply the distributive property.

$\frac{4}{9} (\frac{\tan (t) \sec^{3} (t)}{4} + \frac{3}{4} (\sec (t) \tan (t) - - (\ln (| \sec (t) + \tan (t) |) + C) - \int \sec^{3} (t) d t))$

Step 29.2

Multiply $- 1$ by $- 1$ .

$\frac{4}{9} (\frac{\tan (t) \sec^{3} (t)}{4} + \frac{3}{4} (\sec (t) \tan (t) + 1 (\ln (| \sec (t) + \tan (t) |) + C) - \int \sec^{3} (t) d t))$

Step 30

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

$\frac{4}{9} (\frac{\tan (t) \sec^{3} (t)}{4} + \frac{3}{4} (\frac{\sec (t) \tan (t) + 1 (\ln (| \sec (t) + \tan (t) |) + C)}{2} + C))$

Step 31

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

$\frac{4}{9} (\frac{\tan (t) \sec^{3} (t)}{4} + \frac{3}{4} (\frac{\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |) + C}{2} + C))$

Step 32

Simplify.

$\frac{4}{9} (\frac{\tan (t) \sec^{3} (t)}{4} + \frac{3 (\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |))}{8}) + C$

Step 33

Simplify.

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Step 33.1

To write $\frac{\tan (t) \sec^{3} (t)}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{4}{9} (\frac{\tan (t) \sec^{3} (t)}{4} \cdot \frac{2}{2} + \frac{3 (\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |))}{8}) + C$

Step 33.2

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 33.2.1

Multiply $\frac{\tan (t) \sec^{3} (t)}{4}$ by $\frac{2}{2}$ .

$\frac{4}{9} (\frac{\tan (t) \sec^{3} (t) \cdot 2}{4 \cdot 2} + \frac{3 (\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |))}{8}) + C$

Step 33.2.2

Multiply $4$ by $2$ .

$\frac{4}{9} (\frac{\tan (t) \sec^{3} (t) \cdot 2}{8} + \frac{3 (\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |))}{8}) + C$

Step 33.3

Combine the numerators over the common denominator.

$\frac{4}{9} \cdot \frac{\tan (t) \sec^{3} (t) \cdot 2 + 3 (\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |))}{8} + C$

Step 33.4

Move $2$ to the left of $\tan (t) \sec^{3} (t)$ .

$\frac{4}{9} \cdot \frac{2 \cdot (\tan (t) \sec^{3} (t)) + 3 (\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |))}{8} + C$

Step 33.5

Multiply $\frac{4}{9}$ by $\frac{2 \tan (t) \sec^{3} (t) + 3 (\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |))}{8}$ .

$\frac{4 (2 \tan (t) \sec^{3} (t) + 3 (\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)))}{9 \cdot 8} + C$

Step 33.6

Multiply $9$ by $8$ .

$\frac{4 (2 \tan (t) \sec^{3} (t) + 3 (\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)))}{72} + C$

Step 33.7

Cancel the common factors.

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Step 33.7.1

Factor $4$ out of $72$ .

$\frac{4 (2 \tan (t) \sec^{3} (t) + 3 (\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)))}{4 \cdot 18} + C$

Step 33.7.2

Cancel the common factor.

$\frac{4 (2 \tan (t) \sec^{3} (t) + 3 (\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)))}{4 \cdot 18} + C$

Step 33.7.3

Rewrite the expression.

$\frac{2 \tan (t) \sec^{3} (t) + 3 (\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |))}{18} + C$

Step 34

Replace all occurrences of $t$ with $\arctan (\frac{3 x}{\sqrt{6}})$ .

$\frac{2 \tan (\arctan (\frac{3 x}{\sqrt{6}})) \sec^{3} (\arctan (\frac{3 x}{\sqrt{6}})) + 3 (\sec (\arctan (\frac{3 x}{\sqrt{6}})) \tan (\arctan (\frac{3 x}{\sqrt{6}})) + \ln (| \sec (\arctan (\frac{3 x}{\sqrt{6}})) + \tan (\arctan (\frac{3 x}{\sqrt{6}})) |))}{18} + C$

Step 35

Reorder terms.

$\frac{1}{18} (2 \tan (\arctan (\frac{3}{\sqrt{6}} x)) \sec^{3} (\arctan (\frac{3}{\sqrt{6}} x)) + 3 (\sec (\arctan (\frac{3}{\sqrt{6}} x)) \tan (\arctan (\frac{3}{\sqrt{6}} x)) + \ln (| \sec (\arctan (\frac{3}{\sqrt{6}} x)) + \tan (\arctan (\frac{3}{\sqrt{6}} x)) |))) + C$

Step 36

The answer is the antiderivative of the function $f (x) = \frac{1}{27} \cdot {(9 x^{2} + 6)}^{\frac{3}{2}}$ .

$F (x) =$ $\frac{1}{18} (2 \tan (\arctan (\frac{3}{\sqrt{6}} x)) \sec^{3} (\arctan (\frac{3}{\sqrt{6}} x)) + 3 (\sec (\arctan (\frac{3}{\sqrt{6}} x)) \tan (\arctan (\frac{3}{\sqrt{6}} x)) + \ln (| \sec (\arctan (\frac{3}{\sqrt{6}} x)) + \tan (\arctan (\frac{3}{\sqrt{6}} x)) |))) + C$