Algebra Examples

{sec}^{4} (x)

$\sec^{4} (x)$

Step 1

Simplify the expression.

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Step 1.1

Rewrite

4

$4$ as

2

$2$ plus

2

$2$

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

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

Step 1.2

Rewrite

{sec (x)}^{2 + 2}

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

{sec}^{2} (x) {sec}^{2} (x)

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

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

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

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

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

Step 2

Using the Pythagorean Identity, rewrite

{sec}^{2} (x)

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

1 + {tan}^{2} (x)

$1 + \tan^{2} (x)$ .

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

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

Step 3

Let

u = tan (x)

$u = \tan (x)$ . Then

d u = {sec}^{2} (x) d x

$d u = \sec^{2} (x) d x$ , so

\frac{1}{{sec}^{2} (x)} d u = d x

$\frac{1}{\sec^{2} (x)} d u = d x$ . Rewrite using

u

$u$ and

d

$d$

u

$u$ .

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Step 3.1

Let

u = tan (x)

$u = \tan (x)$ . Find

\frac{d u}{d x}

$\frac{d u}{d x}$ .

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Step 3.1.1

Differentiate

tan (x)

$\tan (x)$ .

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

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

Step 3.1.2

The derivative of

tan (x)

$\tan (x)$ with respect to

x

$x$ is

{sec}^{2} (x)

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

{sec}^{2} (x)

$\sec^{2} (x)$

{sec}^{2} (x)

$\sec^{2} (x)$

Step 3.2

Rewrite the problem using

u

$u$ and

d u

$d u$ .

\int 1 + u^{2} d u

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

\int 1 + u^{2} d u

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

Step 4

Split the single integral into multiple integrals.

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

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

Step 5

Apply the constant rule.

u + C + \int u^{2} d u

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

Step 6

By the Power Rule, the integral of

u^{2}

$u^{2}$ with respect to

u

$u$ is

\frac{1}{3} u^{3}

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

u + C + \frac{1}{3} u^{3} + C

$u + C + \frac{1}{3} u^{3} + C$

Step 7

Simplify.

u + \frac{1}{3} u^{3} + C

$u + \frac{1}{3} u^{3} + C$

Step 8

Replace all occurrences of

u

$u$ with

tan (x)

$\tan (x)$ .

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

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

⎡ ⎢ ⎣ \begin{matrix} x^{2} & \frac{1}{2} \sqrt{π} & \int x d x \end{matrix} ⎤ ⎥ ⎦

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