Calculus Examples

$\int 16 \tan^{4} (x) \sec^{6} (x) d x$

Step 1

Since $16$ is constant with respect to $x$ , move $16$ out of the integral.

$16 \int \tan^{4} (x) \sec^{6} (x) d x$

Step 2

Simplify with factoring out.

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Step 2.1

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

$16 \int \tan^{4} (x) {\sec (x)}^{4 + 2} d x$

Step 2.2

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

$16 \int \tan^{4} (x) (\sec^{4} (x) \sec^{2} (x)) d x$

Step 2.3

Factor $2$ out of $4$ .

$16 \int \tan^{4} (x) ({\sec (x)}^{2 (2)} \sec^{2} (x)) d x$

Step 2.4

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

$16 \int \tan^{4} (x) ({(\sec^{2} (x))}^{2} \sec^{2} (x)) d x$

$16 \int \tan^{4} (x) ({(\sec^{2} (x))}^{2} \sec^{2} (x)) d x$

Step 3

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

$16 \int \tan^{4} (x) ({(1 + \tan^{2} (x))}^{2} \sec^{2} (x)) d x$

Step 4

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 4.1

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

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Step 4.1.1

Differentiate $\tan (x)$ .

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

Step 4.1.2

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

$\sec^{2} (x)$

$\sec^{2} (x)$

Step 4.2

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

$16 \int u^{4} {(1 + u^{2})}^{2} d u$

$16 \int u^{4} {(1 + u^{2})}^{2} d u$

Step 5

Expand $u^{4} {(1 + u^{2})}^{2}$ .

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Step 5.1

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

$16 \int u^{4} ((1 + u^{2}) (1 + u^{2})) d u$

Step 5.2

Apply the distributive property.

$16 \int u^{4} (1 (1 + u^{2}) + u^{2} (1 + u^{2})) d u$

Step 5.3

Apply the distributive property.

$16 \int u^{4} (1 \cdot 1 + 1 u^{2} + u^{2} (1 + u^{2})) d u$

Step 5.4

Apply the distributive property.

$16 \int u^{4} (1 \cdot 1 + 1 u^{2} + u^{2} \cdot 1 + u^{2} u^{2}) d u$

Step 5.5

Apply the distributive property.

$16 \int u^{4} (1 \cdot 1 + 1 u^{2}) + u^{4} (u^{2} \cdot 1 + u^{2} u^{2}) d u$

Step 5.6

Apply the distributive property.

$16 \int u^{4} (1 \cdot 1) + u^{4} (1 u^{2}) + u^{4} (u^{2} \cdot 1 + u^{2} u^{2}) d u$

Step 5.7

Apply the distributive property.

$16 \int u^{4} (1 \cdot 1) + u^{4} (1 u^{2}) + u^{4} (u^{2} \cdot 1) + u^{4} (u^{2} u^{2}) d u$

Step 5.8

Move $u^{4}$ .

$16 \int 1 \cdot 1 u^{4} + u^{4} (1 u^{2}) + u^{4} (u^{2} \cdot 1) + u^{4} (u^{2} u^{2}) d u$

Step 5.9

Reorder $u^{4}$ and $1$ .

$16 \int 1 \cdot 1 u^{4} + 1 \cdot u^{4} u^{2} + u^{4} (u^{2} \cdot 1) + u^{4} (u^{2} u^{2}) d u$

Step 5.10

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

$16 \int 1 \cdot 1 u^{4} + 1 \cdot u^{4} u^{2} + u^{4} (1 \cdot u^{2}) + u^{4} (u^{2} u^{2}) d u$

Step 5.11

Reorder $u^{4}$ and $1$ .

$16 \int 1 \cdot 1 u^{4} + 1 \cdot u^{4} u^{2} + 1 \cdot u^{4} \cdot u^{2} + u^{4} (u^{2} u^{2}) d u$

Step 5.12

Multiply $1$ by $1$ .

$16 \int 1 u^{4} + 1 u^{4} u^{2} + 1 u^{4} u^{2} + u^{4} u^{2} u^{2} d u$

Step 5.13

Multiply $u^{4}$ by $1$ .

$16 \int u^{4} + 1 u^{4} u^{2} + 1 u^{4} u^{2} + u^{4} u^{2} u^{2} d u$

Step 5.14

Multiply $u^{4}$ by $1$ .

$16 \int u^{4} + u^{4} u^{2} + 1 u^{4} u^{2} + u^{4} u^{2} u^{2} d u$

Step 5.15

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

$16 \int u^{4} + u^{4 + 2} + 1 u^{4} u^{2} + u^{4} u^{2} u^{2} d u$

Step 5.16

Add $4$ and $2$ .

$16 \int u^{4} + u^{6} + 1 u^{4} u^{2} + u^{4} u^{2} u^{2} d u$

Step 5.17

Multiply $u^{4}$ by $1$ .

$16 \int u^{4} + u^{6} + u^{4} u^{2} + u^{4} u^{2} u^{2} d u$

Step 5.18

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

$16 \int u^{4} + u^{6} + u^{4 + 2} + u^{4} u^{2} u^{2} d u$

Step 5.19

Add $4$ and $2$ .

$16 \int u^{4} + u^{6} + u^{6} + u^{4} u^{2} u^{2} d u$

Step 5.20

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

$16 \int u^{4} + u^{6} + u^{6} + u^{4 + 2} u^{2} d u$

Step 5.21

Add $4$ and $2$ .

$16 \int u^{4} + u^{6} + u^{6} + u^{6} u^{2} d u$

Step 5.22

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

$16 \int u^{4} + u^{6} + u^{6} + u^{6 + 2} d u$

Step 5.23

Add $6$ and $2$ .

$16 \int u^{4} + u^{6} + u^{6} + u^{8} d u$

Step 5.24

Add $u^{6}$ and $u^{6}$ .

$16 \int u^{4} + 2 u^{6} + u^{8} d u$

Step 5.25

Reorder $2 u^{6}$ and $u^{8}$ .

$16 \int u^{4} + u^{8} + 2 u^{6} d u$

Step 5.26

Move $u^{4}$ .

$16 \int u^{8} + 2 u^{6} + u^{4} d u$

$16 \int u^{8} + 2 u^{6} + u^{4} d u$

Step 6

Split the single integral into multiple integrals.

$16 (\int u^{8} d u + \int 2 u^{6} d u + \int u^{4} d u)$

Step 7

By the Power Rule, the integral of $u^{8}$ with respect to $u$ is $\frac{1}{9} u^{9}$ .

$16 (\frac{1}{9} u^{9} + C + \int 2 u^{6} d u + \int u^{4} d u)$

Step 8

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

$16 (\frac{1}{9} u^{9} + C + 2 \int u^{6} d u + \int u^{4} d u)$

Step 9

By the Power Rule, the integral of $u^{6}$ with respect to $u$ is $\frac{1}{7} u^{7}$ .

$16 (\frac{1}{9} u^{9} + C + 2 (\frac{1}{7} u^{7} + C) + \int u^{4} d u)$

Step 10

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

$16 (\frac{1}{9} u^{9} + C + 2 (\frac{1}{7} u^{7} + C) + \frac{1}{5} u^{5} + C)$

Step 11

Simplify.

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Step 11.1

Simplify.

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Step 11.1.1

Combine $\frac{1}{7}$ and $u^{7}$ .

$16 (\frac{1}{9} u^{9} + C + 2 (\frac{u^{7}}{7} + C) + \frac{1}{5} u^{5} + C)$

Step 11.1.2

Combine $\frac{1}{9}$ and $u^{9}$ .

$16 (\frac{u^{9}}{9} + C + 2 (\frac{u^{7}}{7} + C) + \frac{1}{5} u^{5} + C)$

Step 11.1.3

Combine $\frac{1}{5}$ and $u^{5}$ .

$16 (\frac{u^{9}}{9} + C + 2 (\frac{u^{7}}{7} + C) + \frac{u^{5}}{5} + C)$

$16 (\frac{u^{9}}{9} + C + 2 (\frac{u^{7}}{7} + C) + \frac{u^{5}}{5} + C)$

Step 11.2

Simplify.

$16 (\frac{u^{9}}{9} + \frac{2 u^{7}}{7} + \frac{u^{5}}{5}) + C$

$16 (\frac{u^{9}}{9} + \frac{2 u^{7}}{7} + \frac{u^{5}}{5}) + C$

Step 12

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

$16 (\frac{\tan^{9} (x)}{9} + \frac{2 \tan^{7} (x)}{7} + \frac{\tan^{5} (x)}{5}) + C$

Step 13

Reorder terms.

$16 (\frac{1}{9} \tan^{9} (x) + \frac{2}{7} \tan^{7} (x) + \frac{1}{5} \tan^{5} (x)) + C$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$