Calculus Examples

$\int \tan^{3} (x) \sec^{6} (x) d x$

Step 1

Simplify with factoring out.

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Step 1.1

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

$\int \tan^{3} (x) {\sec (x)}^{4 + 2} d x$

Step 1.2

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

$\int \tan^{3} (x) (\sec^{4} (x) \sec^{2} (x)) d x$

Step 1.3

Factor $2$ out of $4$ .

$\int \tan^{3} (x) ({\sec (x)}^{2 (2)} \sec^{2} (x)) d x$

Step 1.4

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

$\int \tan^{3} (x) ({(\sec^{2} (x))}^{2} \sec^{2} (x)) d x$

$\int \tan^{3} (x) ({(\sec^{2} (x))}^{2} \sec^{2} (x)) d x$

Step 2

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

$\int \tan^{3} (x) ({(1 + \tan^{2} (x))}^{2} \sec^{2} (x)) d x$

Step 3

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 3.1

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

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Step 3.1.1

Differentiate $\tan (x)$ .

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

Step 3.1.2

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

$\sec^{2} (x)$

$\sec^{2} (x)$

Step 3.2

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

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

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

Step 4

Expand $u^{3} {(1 + u^{2})}^{2}$ .

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Step 4.1

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

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

$\int u^{3} (1 \cdot 1 + 1 u^{2} + u^{2} (1 + u^{2})) d u$

Step 4.4

Apply the distributive property.

$\int u^{3} (1 \cdot 1 + 1 u^{2} + u^{2} \cdot 1 + u^{2} u^{2}) d u$

Step 4.5

Apply the distributive property.

$\int u^{3} (1 \cdot 1 + 1 u^{2}) + u^{3} (u^{2} \cdot 1 + u^{2} u^{2}) d u$

Step 4.6

Apply the distributive property.

$\int u^{3} (1 \cdot 1) + u^{3} (1 u^{2}) + u^{3} (u^{2} \cdot 1 + u^{2} u^{2}) d u$

Step 4.7

Apply the distributive property.

$\int u^{3} (1 \cdot 1) + u^{3} (1 u^{2}) + u^{3} (u^{2} \cdot 1) + u^{3} (u^{2} u^{2}) d u$

Step 4.8

Move $u^{3}$ .

$\int 1 \cdot 1 u^{3} + u^{3} (1 u^{2}) + u^{3} (u^{2} \cdot 1) + u^{3} (u^{2} u^{2}) d u$

Step 4.9

Reorder $u^{3}$ and $1$ .

$\int 1 \cdot 1 u^{3} + 1 \cdot u^{3} u^{2} + u^{3} (u^{2} \cdot 1) + u^{3} (u^{2} u^{2}) d u$

Step 4.10

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

$\int 1 \cdot 1 u^{3} + 1 \cdot u^{3} u^{2} + u^{3} (1 \cdot u^{2}) + u^{3} (u^{2} u^{2}) d u$

Step 4.11

Reorder $u^{3}$ and $1$ .

$\int 1 \cdot 1 u^{3} + 1 \cdot u^{3} u^{2} + 1 \cdot u^{3} \cdot u^{2} + u^{3} (u^{2} u^{2}) d u$

Step 4.12

Multiply $1$ by $1$ .

$\int 1 u^{3} + 1 u^{3} u^{2} + 1 u^{3} u^{2} + u^{3} u^{2} u^{2} d u$

Step 4.13

Multiply $u^{3}$ by $1$ .

$\int u^{3} + 1 u^{3} u^{2} + 1 u^{3} u^{2} + u^{3} u^{2} u^{2} d u$

Step 4.14

Multiply $u^{3}$ by $1$ .

$\int u^{3} + u^{3} u^{2} + 1 u^{3} u^{2} + u^{3} u^{2} u^{2} d u$

Step 4.15

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

$\int u^{3} + u^{3 + 2} + 1 u^{3} u^{2} + u^{3} u^{2} u^{2} d u$

Step 4.16

Add $3$ and $2$ .

$\int u^{3} + u^{5} + 1 u^{3} u^{2} + u^{3} u^{2} u^{2} d u$

Step 4.17

Multiply $u^{3}$ by $1$ .

$\int u^{3} + u^{5} + u^{3} u^{2} + u^{3} u^{2} u^{2} d u$

Step 4.18

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

$\int u^{3} + u^{5} + u^{3 + 2} + u^{3} u^{2} u^{2} d u$

Step 4.19

Add $3$ and $2$ .

$\int u^{3} + u^{5} + u^{5} + u^{3} u^{2} u^{2} d u$

Step 4.20

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

$\int u^{3} + u^{5} + u^{5} + u^{3 + 2} u^{2} d u$

Step 4.21

Add $3$ and $2$ .

$\int u^{3} + u^{5} + u^{5} + u^{5} u^{2} d u$

Step 4.22

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

$\int u^{3} + u^{5} + u^{5} + u^{5 + 2} d u$

Step 4.23

Add $5$ and $2$ .

$\int u^{3} + u^{5} + u^{5} + u^{7} d u$

Step 4.24

Add $u^{5}$ and $u^{5}$ .

$\int u^{3} + 2 u^{5} + u^{7} d u$

Step 4.25

Reorder $2 u^{5}$ and $u^{7}$ .

$\int u^{3} + u^{7} + 2 u^{5} d u$

Step 4.26

Move $u^{3}$ .

$\int u^{7} + 2 u^{5} + u^{3} d u$

$\int u^{7} + 2 u^{5} + u^{3} d u$

Step 5

Split the single integral into multiple integrals.

$\int u^{7} d u + \int 2 u^{5} d u + \int u^{3} d u$

Step 6

By the Power Rule, the integral of $u^{7}$ with respect to $u$ is $\frac{1}{8} u^{8}$ .

$\frac{1}{8} u^{8} + C + \int 2 u^{5} d u + \int u^{3} d u$

Step 7

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

$\frac{1}{8} u^{8} + C + 2 \int u^{5} d u + \int u^{3} d u$

Step 8

By the Power Rule, the integral of $u^{5}$ with respect to $u$ is $\frac{1}{6} u^{6}$ .

$\frac{1}{8} u^{8} + C + 2 (\frac{1}{6} u^{6} + C) + \int u^{3} d u$

Step 9

By the Power Rule, the integral of $u^{3}$ with respect to $u$ is $\frac{1}{4} u^{4}$ .

$\frac{1}{8} u^{8} + C + 2 (\frac{1}{6} u^{6} + C) + \frac{1}{4} u^{4} + C$

Step 10

Simplify.

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Step 10.1

Combine $\frac{1}{6}$ and $u^{6}$ .

$\frac{1}{8} u^{8} + C + 2 (\frac{u^{6}}{6} + C) + \frac{1}{4} u^{4} + C$

Step 10.2

Simplify.

$\frac{u^{8}}{8} + \frac{u^{6}}{3} + \frac{1}{4} u^{4} + C$

$\frac{u^{8}}{8} + \frac{u^{6}}{3} + \frac{1}{4} u^{4} + C$

Step 11

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

$\frac{\tan^{8} (x)}{8} + \frac{\tan^{6} (x)}{3} + \frac{1}{4} \tan^{4} (x) + C$

Step 12

Reorder terms.

$\frac{1}{8} \tan^{8} (x) + \frac{1}{3} \tan^{6} (x) + \frac{1}{4} \tan^{4} (x) + C$

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