Calculus Examples

$\sec^{6} (x)$

Step 1

Write $\sec^{6} (x)$ as a function.

$f (x) = \sec^{6} (x)$

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.

$F (x) = \int \sec^{6} (x) d x$

Step 4

Simplify with factoring out.

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Step 4.1

Rewrite $6$ as $4$ plus $2$

$\int {\sec (x)}^{4 + 2} d x$

Step 4.2

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

$\int \sec^{4} (x) \sec^{2} (x) d x$

Step 4.3

Factor $2$ out of $4$ .

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

Step 4.4

Rewrite ${\sec (x)}^{2 (2)}$ as exponentiation.

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

Step 5

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

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

Step 6

Let $u = \tan (x)$ . Then $d u = \sec^{2} (x) d x$ , so $\frac{1}{\sec^{2} (x)} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.1

Let $u = \tan (x)$ . Find $\frac{d u}{d x}$ .

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Step 6.1.1

Differentiate $\tan (x)$ .

$\frac{d}{d x} [\tan (x)]$

Step 6.1.2

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

$\sec^{2} (x)$

Step 6.2

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

$\int {(1 + u^{2})}^{2} d u$

Step 7

Expand ${(1 + u^{2})}^{2}$ .

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Step 7.1

Rewrite ${(1 + u^{2})}^{2}$ as $(1 + u^{2}) (1 + u^{2})$ .

$\int (1 + u^{2}) (1 + u^{2}) d u$

Step 7.2

Apply the distributive property.

$\int 1 (1 + u^{2}) + u^{2} (1 + u^{2}) d u$

Step 7.3

Apply the distributive property.

$\int 1 \cdot 1 + 1 u^{2} + u^{2} (1 + u^{2}) d u$

Step 7.4

Apply the distributive property.

$\int 1 \cdot 1 + 1 u^{2} + u^{2} \cdot 1 + u^{2} u^{2} d u$

Step 7.5

Reorder $u^{2}$ and $1$ .

$\int 1 \cdot 1 + 1 u^{2} + 1 \cdot u^{2} + u^{2} u^{2} d u$

Step 7.6

Multiply $1$ by $1$ .

$\int 1 + 1 u^{2} + 1 u^{2} + u^{2} u^{2} d u$

Step 7.7

Multiply $u^{2}$ by $1$ .

$\int 1 + u^{2} + 1 u^{2} + u^{2} u^{2} d u$

Step 7.8

Multiply $u^{2}$ by $1$ .

$\int 1 + u^{2} + u^{2} + u^{2} u^{2} d u$

Step 7.9

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

$\int 1 + u^{2} + u^{2} + u^{2 + 2} d u$

Step 7.10

Add $2$ and $2$ .

$\int 1 + u^{2} + u^{2} + u^{4} d u$

Step 7.11

Add $u^{2}$ and $u^{2}$ .

$\int 1 + 2 u^{2} + u^{4} d u$

Step 7.12

Reorder $2 u^{2}$ and $u^{4}$ .

$\int 1 + u^{4} + 2 u^{2} d u$

Step 7.13

Move $1$ .

$\int u^{4} + 2 u^{2} + 1 d u$

Step 8

Split the single integral into multiple integrals.

$\int u^{4} d u + \int 2 u^{2} d u + \int d u$

Step 9

By the Power Rule, the integral of $u^{4}$ with respect to $u$ is $\frac{1}{5} u^{5}$ .

$\frac{1}{5} u^{5} + C + \int 2 u^{2} d u + \int d u$

Step 10

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

$\frac{1}{5} u^{5} + C + 2 \int u^{2} d u + \int d u$

Step 11

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

$\frac{1}{5} u^{5} + C + 2 (\frac{1}{3} u^{3} + C) + \int d u$

Step 12

Apply the constant rule.

$\frac{1}{5} u^{5} + C + 2 (\frac{1}{3} u^{3} + C) + u + C$

Step 13

Simplify.

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Step 13.1

Combine $\frac{1}{3}$ and $u^{3}$ .

$\frac{1}{5} u^{5} + C + 2 (\frac{u^{3}}{3} + C) + u + C$

Step 13.2

Simplify.

$\frac{u^{5}}{5} + \frac{2 u^{3}}{3} + u + C$

Step 14

Replace all occurrences of $u$ with $\tan (x)$ .

$\frac{\tan^{5} (x)}{5} + \frac{2 \tan^{3} (x)}{3} + \tan (x) + C$

Step 15

Reorder terms.

$\frac{1}{5} \tan^{5} (x) + \frac{2}{3} \tan^{3} (x) + \tan (x) + C$

Step 16

The answer is the antiderivative of the function $f (x) = \sec^{6} (x)$ .

$F (x) =$ $\frac{1}{5} \tan^{5} (x) + \frac{2}{3} \tan^{3} (x) + \tan (x) + C$