Calculus Examples

$\int \sqrt{9 x^{4} - 6 x^{2} + 1} d x$

Step 1

Let $x = \frac{1}{\sqrt{3}} \sec (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\frac{\sqrt{3} \sec (t) \tan (t)}{3}$ is positive.

$\int \sqrt{9 {(\frac{1}{\sqrt{3}} \sec (t))}^{4} - 6 {(\frac{1}{\sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2

Simplify $\sqrt{9 {(\frac{1}{\sqrt{3}} \sec (t))}^{4} - 6 {(\frac{1}{\sqrt{3}} \sec (t))}^{2} + 1}$ .

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\int \sqrt{9 {(\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} \sec (t))}^{4} - 6 {(\frac{1}{\sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.2

Combine and simplify the denominator.

Tap for more steps...

Step 2.1.2.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\int \sqrt{9 {(\frac{\sqrt{3}}{\sqrt{3} \sqrt{3}} \sec (t))}^{4} - 6 {(\frac{1}{\sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.2.2

Raise $\sqrt{3}$ to the power of $1$ .

$\int \sqrt{9 {(\frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} \sec (t))}^{4} - 6 {(\frac{1}{\sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.2.3

Raise $\sqrt{3}$ to the power of $1$ .

$\int \sqrt{9 {(\frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} \sec (t))}^{4} - 6 {(\frac{1}{\sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.2.4

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

$\int \sqrt{9 {(\frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}} \sec (t))}^{4} - 6 {(\frac{1}{\sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.2.5

Add $1$ and $1$ .

$\int \sqrt{9 {(\frac{\sqrt{3}}{{\sqrt{3}}^{2}} \sec (t))}^{4} - 6 {(\frac{1}{\sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.2.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Tap for more steps...

Step 2.1.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\int \sqrt{9 {(\frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} \sec (t))}^{4} - 6 {(\frac{1}{\sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.2.6.2

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

$\int \sqrt{9 {(\frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}} \sec (t))}^{4} - 6 {(\frac{1}{\sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.2.6.3

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

$\int \sqrt{9 {(\frac{\sqrt{3}}{3^{\frac{2}{2}}} \sec (t))}^{4} - 6 {(\frac{1}{\sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.2.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.1.2.6.4.1

Cancel the common factor.

$\int \sqrt{9 {(\frac{\sqrt{3}}{3^{\frac{2}{2}}} \sec (t))}^{4} - 6 {(\frac{1}{\sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.2.6.4.2

Rewrite the expression.

$\int \sqrt{9 {(\frac{\sqrt{3}}{3^{1}} \sec (t))}^{4} - 6 {(\frac{1}{\sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.2.6.5

Evaluate the exponent.

$\int \sqrt{9 {(\frac{\sqrt{3}}{3} \sec (t))}^{4} - 6 {(\frac{1}{\sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.3

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

$\int \sqrt{9 {(\frac{\sqrt{3} \sec (t)}{3})}^{4} - 6 {(\frac{1}{\sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.4

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

Tap for more steps...

Step 2.1.4.1

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

$\int \sqrt{9 \frac{{(\sqrt{3} \sec (t))}^{4}}{3^{4}} - 6 {(\frac{1}{\sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.4.2

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

$\int \sqrt{9 \frac{{\sqrt{3}}^{4} \sec^{4} (t)}{3^{4}} - 6 {(\frac{1}{\sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.5

Simplify the numerator.

Tap for more steps...

Step 2.1.5.1

Rewrite ${\sqrt{3}}^{4}$ as $3^{2}$ .

Tap for more steps...

Step 2.1.5.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\int \sqrt{9 \frac{{(3^{\frac{1}{2}})}^{4} \sec^{4} (t)}{3^{4}} - 6 {(\frac{1}{\sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.5.1.2

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

$\int \sqrt{9 \frac{3^{\frac{1}{2} \cdot 4} \sec^{4} (t)}{3^{4}} - 6 {(\frac{1}{\sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.5.1.3

Combine $\frac{1}{2}$ and $4$ .

$\int \sqrt{9 \frac{3^{\frac{4}{2}} \sec^{4} (t)}{3^{4}} - 6 {(\frac{1}{\sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.5.1.4

Cancel the common factor of $4$ and $2$ .

Tap for more steps...

Step 2.1.5.1.4.1

Factor $2$ out of $4$ .

$\int \sqrt{9 \frac{3^{\frac{2 \cdot 2}{2}} \sec^{4} (t)}{3^{4}} - 6 {(\frac{1}{\sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.5.1.4.2

Cancel the common factors.

Tap for more steps...

Step 2.1.5.1.4.2.1

Factor $2$ out of $2$ .

$\int \sqrt{9 \frac{3^{\frac{2 \cdot 2}{2 (1)}} \sec^{4} (t)}{3^{4}} - 6 {(\frac{1}{\sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.5.1.4.2.2

Cancel the common factor.

$\int \sqrt{9 \frac{3^{\frac{2 \cdot 2}{2 \cdot 1}} \sec^{4} (t)}{3^{4}} - 6 {(\frac{1}{\sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.5.1.4.2.3

Rewrite the expression.

$\int \sqrt{9 \frac{3^{\frac{2}{1}} \sec^{4} (t)}{3^{4}} - 6 {(\frac{1}{\sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.5.1.4.2.4

Divide $2$ by $1$ .

$\int \sqrt{9 \frac{3^{2} \sec^{4} (t)}{3^{4}} - 6 {(\frac{1}{\sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.5.2

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

$\int \sqrt{9 \frac{9 \sec^{4} (t)}{3^{4}} - 6 {(\frac{1}{\sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.6

Raise $3$ to the power of $4$ .

$\int \sqrt{9 \frac{9 \sec^{4} (t)}{81} - 6 {(\frac{1}{\sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.7

Cancel the common factor of $9$ .

Tap for more steps...

Step 2.1.7.1

Factor $9$ out of $81$ .

$\int \sqrt{9 \frac{9 \sec^{4} (t)}{9 (9)} - 6 {(\frac{1}{\sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.7.2

Cancel the common factor.

$\int \sqrt{9 \frac{9 \sec^{4} (t)}{9 \cdot 9} - 6 {(\frac{1}{\sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.7.3

Rewrite the expression.

$\int \sqrt{\frac{9 \sec^{4} (t)}{9} - 6 {(\frac{1}{\sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.8

Cancel the common factor of $9$ .

Tap for more steps...

Step 2.1.8.1

Cancel the common factor.

$\int \sqrt{\frac{9 \sec^{4} (t)}{9} - 6 {(\frac{1}{\sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.8.2

Divide $\sec^{4} (t)$ by $1$ .

$\int \sqrt{\sec^{4} (t) - 6 {(\frac{1}{\sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.9

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\int \sqrt{\sec^{4} (t) - 6 {(\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.10

Combine and simplify the denominator.

Tap for more steps...

Step 2.1.10.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\int \sqrt{\sec^{4} (t) - 6 {(\frac{\sqrt{3}}{\sqrt{3} \sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.10.2

Raise $\sqrt{3}$ to the power of $1$ .

$\int \sqrt{\sec^{4} (t) - 6 {(\frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.10.3

Raise $\sqrt{3}$ to the power of $1$ .

$\int \sqrt{\sec^{4} (t) - 6 {(\frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.10.4

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

$\int \sqrt{\sec^{4} (t) - 6 {(\frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.10.5

Add $1$ and $1$ .

$\int \sqrt{\sec^{4} (t) - 6 {(\frac{\sqrt{3}}{{\sqrt{3}}^{2}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.10.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Tap for more steps...

Step 2.1.10.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\int \sqrt{\sec^{4} (t) - 6 {(\frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.10.6.2

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

$\int \sqrt{\sec^{4} (t) - 6 {(\frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.10.6.3

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

$\int \sqrt{\sec^{4} (t) - 6 {(\frac{\sqrt{3}}{3^{\frac{2}{2}}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.10.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.1.10.6.4.1

Cancel the common factor.

$\int \sqrt{\sec^{4} (t) - 6 {(\frac{\sqrt{3}}{3^{\frac{2}{2}}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.10.6.4.2

Rewrite the expression.

$\int \sqrt{\sec^{4} (t) - 6 {(\frac{\sqrt{3}}{3^{1}} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.10.6.5

Evaluate the exponent.

$\int \sqrt{\sec^{4} (t) - 6 {(\frac{\sqrt{3}}{3} \sec (t))}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.11

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

$\int \sqrt{\sec^{4} (t) - 6 {(\frac{\sqrt{3} \sec (t)}{3})}^{2} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.12

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

Tap for more steps...

Step 2.1.12.1

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

$\int \sqrt{\sec^{4} (t) - 6 \frac{{(\sqrt{3} \sec (t))}^{2}}{3^{2}} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.12.2

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

$\int \sqrt{\sec^{4} (t) - 6 \frac{{\sqrt{3}}^{2} \sec^{2} (t)}{3^{2}} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.13

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Tap for more steps...

Step 2.1.13.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\int \sqrt{\sec^{4} (t) - 6 \frac{{(3^{\frac{1}{2}})}^{2} \sec^{2} (t)}{3^{2}} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.13.2

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

$\int \sqrt{\sec^{4} (t) - 6 \frac{3^{\frac{1}{2} \cdot 2} \sec^{2} (t)}{3^{2}} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.13.3

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

$\int \sqrt{\sec^{4} (t) - 6 \frac{3^{\frac{2}{2}} \sec^{2} (t)}{3^{2}} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.13.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.1.13.4.1

Cancel the common factor.

$\int \sqrt{\sec^{4} (t) - 6 \frac{3^{\frac{2}{2}} \sec^{2} (t)}{3^{2}} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.13.4.2

Rewrite the expression.

$\int \sqrt{\sec^{4} (t) - 6 \frac{3^{1} \sec^{2} (t)}{3^{2}} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.13.5

Evaluate the exponent.

$\int \sqrt{\sec^{4} (t) - 6 \frac{3 \sec^{2} (t)}{3^{2}} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.14

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

$\int \sqrt{\sec^{4} (t) - 6 \frac{3 \sec^{2} (t)}{9} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.15

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.1.15.1

Factor $3$ out of $- 6$ .

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

Step 2.1.15.2

Factor $3$ out of $9$ .

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

Step 2.1.15.3

Cancel the common factor.

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

Step 2.1.15.4

Rewrite the expression.

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

Step 2.1.16

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

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

Step 2.1.17

Multiply $3$ by $- 2$ .

$\int \sqrt{\sec^{4} (t) + \frac{- 6 \sec^{2} (t)}{3} + 1} \frac{\sqrt{3} \sec (t) \tan (t)}{3} d t$

Step 2.1.18

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

Tap for more steps...

Step 2.1.18.1

Factor $3$ out of $- 6 \sec^{2} (t)$ .

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

Step 2.1.18.2

Cancel the common factors.

Tap for more steps...

Step 2.1.18.2.1

Factor $3$ out of $3$ .

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

Step 2.1.18.2.2

Cancel the common factor.

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

Step 2.1.18.2.3

Rewrite the expression.

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

Step 2.1.18.2.4

Divide $- 2 \sec^{2} (t)$ by $1$ .

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

Step 2.2

Factor using the perfect square rule.

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 \sec^{2} (t) = 2 \cdot \sec^{2} (t) \cdot 1$

Step 2.2.4

Rewrite the polynomial.

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

Step 2.2.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = \sec^{2} (t)$ and $b = 1$ .

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

Step 2.3

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

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

Step 3

Let $u = \sec (t)$ . Then $d u = \sec (t) \tan (t) d t$ , so $\frac{1}{\sec (t) \tan (t)} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 3.1

Let $u = \sec (t)$ . Find $\frac{d u}{d t}$ .

Tap for more steps...

Step 3.1.1

Differentiate $\sec (t)$ .

$\frac{d}{d t} [\sec (t)]$

Step 3.1.2

The derivative of $\sec (t)$ with respect to $t$ is $\sec (t) \tan (t)$ .

$\sec (t) \tan (t)$

Step 3.2

Rewrite the problem using $u$ and $d u$ .

$\int (u^{2} - 1) \frac{\sqrt{3}}{3} d u$

Step 4

Multiply $(u^{2} - 1) \frac{\sqrt{3}}{3}$ .

$\int u^{2} \frac{\sqrt{3}}{3} - 1 \frac{\sqrt{3}}{3} d u$

Step 5

Simplify.

Tap for more steps...

Step 5.1

Combine $u^{2}$ and $\frac{\sqrt{3}}{3}$ .

$\int \frac{u^{2} \sqrt{3}}{3} - 1 \frac{\sqrt{3}}{3} d u$

Step 5.2

Rewrite $- 1 \frac{\sqrt{3}}{3}$ as $- \frac{\sqrt{3}}{3}$ .

$\int \frac{u^{2} \sqrt{3}}{3} - \frac{\sqrt{3}}{3} d u$

Step 6

Split the single integral into multiple integrals.

$\int \frac{u^{2} \sqrt{3}}{3} d u + \int - \frac{\sqrt{3}}{3} d u$

Step 7

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

$\frac{\sqrt{3}}{3} \int u^{2} d u + \int - \frac{\sqrt{3}}{3} d u$

Step 8

By the Power Rule, the integral of $u^{2}$ with respect to $u$ is $\frac{1}{3} u^{3}$ .

$\frac{\sqrt{3}}{3} (\frac{1}{3} u^{3} + C) + \int - \frac{\sqrt{3}}{3} d u$

Step 9

Apply the constant rule.

$\frac{\sqrt{3}}{3} (\frac{1}{3} u^{3} + C) - \frac{\sqrt{3}}{3} u + C$

Step 10

Simplify.

Tap for more steps...

Step 10.1

Simplify.

$\frac{\sqrt{3}}{3} \cdot \frac{1}{3} u^{3} - \frac{\sqrt{3}}{3} u + C$

Step 10.2

Simplify.

Tap for more steps...

Step 10.2.1

Multiply $\frac{\sqrt{3}}{3}$ by $\frac{1}{3}$ .

$\frac{\sqrt{3}}{3 \cdot 3} u^{3} - \frac{\sqrt{3}}{3} u + C$

Step 10.2.2

Multiply $3$ by $3$ .

$\frac{\sqrt{3}}{9} u^{3} - \frac{\sqrt{3}}{3} u + C$

Step 11

Substitute back in for each integration substitution variable.

Tap for more steps...

Step 11.1

Replace all occurrences of $u$ with $\sec (t)$ .

$\frac{\sqrt{3}}{9} \sec^{3} (t) - \frac{\sqrt{3}}{3} \sec (t) + C$

Step 11.2

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

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