Calculus Examples

$\sec^{3} (2 x)$

Step 1

Factor $\sec (2 x)$ out of $\sec^{3} (2 x)$ .

$\int \sec (2 x) \sec^{2} (2 x) d x$

Step 2

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

$\sec (2 x) (\frac{1}{2} \tan (2 x)) - \int \frac{1}{2} \tan (2 x) (2 \sec (2 x) \tan (2 x)) d x$

Step 3

Since $\frac{1}{2} \cdot 2$ is constant with respect to $x$ , move $\frac{1}{2} \cdot 2$ out of the integral.

$\sec (2 x) (\frac{1}{2} \tan (2 x)) - (\frac{1}{2} \cdot 2 \int \tan (2 x) (\sec (2 x) \tan (2 x)) d x)$

Step 4

Combine fractions.

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Step 4.1

Combine $\frac{1}{2}$ and $\tan (2 x)$ .

$\sec (2 x) \frac{\tan (2 x)}{2} - (\frac{1}{2} \cdot 2 \int \tan (2 x) (\sec (2 x) \tan (2 x)) d x)$

Step 4.2

Combine $\sec (2 x)$ and $\frac{\tan (2 x)}{2}$ .

$\frac{\sec (2 x) \tan (2 x)}{2} - (\frac{1}{2} \cdot 2 \int \tan (2 x) (\sec (2 x) \tan (2 x)) d x)$

Step 4.3

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

$\frac{\sec (2 x) \tan (2 x)}{2} - (\frac{2}{2} \int \tan (2 x) (\sec (2 x) \tan (2 x)) d x)$

Step 5

Raise $\tan (2 x)$ to the power of $1$ .

$\frac{\sec (2 x) \tan (2 x)}{2} - (\frac{2}{2} \int \tan^{1} (2 x) \tan (2 x) \sec (2 x) d x)$

Step 6

Raise $\tan (2 x)$ to the power of $1$ .

$\frac{\sec (2 x) \tan (2 x)}{2} - (\frac{2}{2} \int \tan^{1} (2 x) \tan^{1} (2 x) \sec (2 x) d x)$

Step 7

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

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

Step 8

Simplify the expression.

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Step 8.1

Add $1$ and $1$ .

$\frac{\sec (2 x) \tan (2 x)}{2} - (\frac{2}{2} \int \tan^{2} (2 x) \sec (2 x) d x)$

Step 8.2

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

$\frac{\sec (2 x) \tan (2 x)}{2} - (\frac{2}{2} \int \sec (2 x) \tan^{2} (2 x) d x)$

Step 9

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

$\frac{\sec (2 x) \tan (2 x)}{2} - (\frac{2}{2} (\sec (2 x) (- x + \frac{1}{2} \tan (2 x)) - \int (- x + \frac{1}{2} \tan (2 x)) (2 \sec (2 x) \tan (2 x)) d x))$

Step 10

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$\frac{\sec (2 x) \tan (2 x)}{2} - (\frac{2}{2} (\sec (2 x) (- x + \frac{1}{2} \tan (2 x)) - (2 \int (- x + \frac{1}{2} \tan (2 x)) (\sec (2 x) \tan (2 x)) d x)))$

Step 11

Simplify terms.

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Step 11.1

Combine $\frac{1}{2}$ and $\tan (2 x)$ .

$\frac{\sec (2 x) \tan (2 x)}{2} - (\frac{2}{2} (\sec (2 x) (- x + \frac{\tan (2 x)}{2}) - (2 \int (- x + \frac{1}{2} \tan (2 x)) (\sec (2 x) \tan (2 x)) d x)))$

Step 11.2

Combine $\frac{1}{2}$ and $\tan (2 x)$ .

$\frac{\sec (2 x) \tan (2 x)}{2} - (\frac{2}{2} (\sec (2 x) (- x + \frac{\tan (2 x)}{2}) - (2 \int (- x + \frac{\tan (2 x)}{2}) (\sec (2 x) \tan (2 x)) d x)))$

Step 11.3

Multiply $2$ by $- 1$ .

$\frac{\sec (2 x) \tan (2 x)}{2} - (\frac{2}{2} (\sec (2 x) (- x + \frac{\tan (2 x)}{2}) - 2 \int (- x + \frac{\tan (2 x)}{2}) (\sec (2 x) \tan (2 x)) d x))$

Step 11.4

Apply the distributive property.

$\frac{\sec (2 x) \tan (2 x)}{2} - \frac{2}{2} (\sec (2 x) (- x + \frac{\tan (2 x)}{2})) - \frac{2}{2} (- 2 \int (- x + \frac{\tan (2 x)}{2}) \sec (2 x) \tan (2 x) d x)$

Step 11.5

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

$\frac{\sec (2 x) \tan (2 x)}{2} - \frac{\sec (2 x) \cdot 2}{2} (- x + \frac{\tan (2 x)}{2}) - \frac{2}{2} (- 2 \int (- x + \frac{\tan (2 x)}{2}) \sec (2 x) \tan (2 x) d x)$

Step 11.6

Multiply $- 2$ by $- 1$ .

$\frac{\sec (2 x) \tan (2 x)}{2} - \frac{\sec (2 x) \cdot 2}{2} (- x + \frac{\tan (2 x)}{2}) + 2 (\frac{2}{2}) \int (- x + \frac{\tan (2 x)}{2}) \sec (2 x) \tan (2 x) d x$

Step 11.7

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

$\frac{\sec (2 x) \tan (2 x)}{2} - \frac{\sec (2 x) \cdot 2}{2} (- x + \frac{\tan (2 x)}{2}) + \frac{2 \cdot 2}{2} \int (- x + \frac{\tan (2 x)}{2}) \sec (2 x) \tan (2 x) d x$

Step 11.8

Multiply $2$ by $2$ .

$\frac{\sec (2 x) \tan (2 x)}{2} - \frac{\sec (2 x) \cdot 2}{2} (- x + \frac{\tan (2 x)}{2}) + \frac{4}{2} \int (- x + \frac{\tan (2 x)}{2}) \sec (2 x) \tan (2 x) d x$

Step 12

Solving for $\int \sec^{3} (2 x) d x$ , we find that $\int \sec^{3} (2 x) d x$ = $\frac{\frac{\sec (2 x) \tan (2 x)}{2} - \frac{\sec (2 x) \cdot 2}{2} (- x + \frac{\tan (2 x)}{2})}{\frac{- 2}{2}}$ .

$\frac{\frac{\sec (2 x) \tan (2 x)}{2} - \frac{\sec (2 x) \cdot 2}{2} (- x + \frac{\tan (2 x)}{2})}{\frac{- 2}{2}} + C$

Step 13

Simplify the answer.

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Step 13.1

Simplify.

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Step 13.1.1

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

$\frac{\frac{\sec (2 x) \tan (2 x)}{2} - \frac{2 \cdot \sec (2 x)}{2} (- x + \frac{\tan (2 x)}{2})}{\frac{- 2}{2}} + C$

Step 13.1.2

Cancel the common factor of $2$ .

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Step 13.1.2.1

Cancel the common factor.

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

Step 13.1.2.2

Divide $\sec (2 x)$ by $1$ .

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

Step 13.1.3

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

$\frac{\frac{\sec (2 x) \tan (2 x)}{2} - \sec (2 x) (- x + \frac{\tan (2 x)}{2}) \cdot \frac{2}{2}}{\frac{- 2}{2}} + C$

Step 13.1.4

Combine $- \sec (2 x) (- x + \frac{\tan (2 x)}{2})$ and $\frac{2}{2}$ .

$\frac{\frac{\sec (2 x) \tan (2 x)}{2} + \frac{- \sec (2 x) (- x + \frac{\tan (2 x)}{2}) \cdot 2}{2}}{\frac{- 2}{2}} + C$

Step 13.1.5

Combine the numerators over the common denominator.

$\frac{\frac{\sec (2 x) \tan (2 x) - \sec (2 x) (- x + \frac{\tan (2 x)}{2}) \cdot 2}{2}}{\frac{- 2}{2}} + C$

Step 13.1.6

Multiply $2$ by $- 1$ .

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

Step 13.1.7

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

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Step 13.1.7.1

Factor $2$ out of $- 2$ .

$\frac{\frac{\sec (2 x) \tan (2 x) - 2 \sec (2 x) (- x + \frac{\tan (2 x)}{2})}{2}}{\frac{2 \cdot - 1}{2}} + C$

Step 13.1.7.2

Cancel the common factors.

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Step 13.1.7.2.1

Factor $2$ out of $2$ .

$\frac{\frac{\sec (2 x) \tan (2 x) - 2 \sec (2 x) (- x + \frac{\tan (2 x)}{2})}{2}}{\frac{2 \cdot - 1}{2 (1)}} + C$

Step 13.1.7.2.2

Cancel the common factor.

$\frac{\frac{\sec (2 x) \tan (2 x) - 2 \sec (2 x) (- x + \frac{\tan (2 x)}{2})}{2}}{\frac{2 \cdot - 1}{2 \cdot 1}} + C$

Step 13.1.7.2.3

Rewrite the expression.

$\frac{\frac{\sec (2 x) \tan (2 x) - 2 \sec (2 x) (- x + \frac{\tan (2 x)}{2})}{2}}{\frac{- 1}{1}} + C$

Step 13.1.7.2.4

Divide $- 1$ by $1$ .

$\frac{\frac{\sec (2 x) \tan (2 x) - 2 \sec (2 x) (- x + \frac{\tan (2 x)}{2})}{2}}{- 1} + C$

Step 13.1.8

Move the negative one from the denominator of $\frac{\frac{\sec (2 x) \tan (2 x) - 2 \sec (2 x) (- x + \frac{\tan (2 x)}{2})}{2}}{- 1}$ .

$- 1 \cdot \frac{\sec (2 x) \tan (2 x) - 2 \sec (2 x) (- x + \frac{\tan (2 x)}{2})}{2} + C$

Step 13.1.9

Rewrite $- 1 \frac{\sec (2 x) \tan (2 x) - 2 \sec (2 x) (- x + \frac{\tan (2 x)}{2})}{2}$ as $- \frac{\sec (2 x) \tan (2 x) - 2 \sec (2 x) (- x + \frac{\tan (2 x)}{2})}{2}$ .

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

Step 13.2

Rewrite $- \frac{\sec (2 x) \tan (2 x) - 2 \sec (2 x) (- x + \frac{\tan (2 x)}{2})}{2} + C$ as $- \frac{\sec (2 x) \cdot 2 x}{2} + C$ .

$- \frac{\sec (2 x) \cdot 2 x}{2} + C$

Step 13.3

Simplify.

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Step 13.3.1

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

$- \frac{2 \cdot \sec (2 x) x}{2} + C$

Step 13.3.2

Cancel the common factor of $2$ .

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Step 13.3.2.1

Cancel the common factor.

$- \frac{2 \sec (2 x) x}{2} + C$

Step 13.3.2.2

Divide $\sec (2 x) x$ by $1$ .

$- (\sec (2 x) x) + C$

$- \sec (2 x) x + C$