Calculus Examples

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

Step 1

Write ${(\sec (x) + \tan (x))}^{2}$ as a function.

$f (x) = {(\sec (x) + \tan (x))}^{2}$

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.

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

Step 4

Simplify.

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Step 4.1

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

$\int (\sec (x) + \tan (x)) (\sec (x) + \tan (x)) d x$

Step 4.2

Expand $(\sec (x) + \tan (x)) (\sec (x) + \tan (x))$ using the FOIL Method.

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Step 4.2.1

Apply the distributive property.

$\int \sec (x) (\sec (x) + \tan (x)) + \tan (x) (\sec (x) + \tan (x)) d x$

Step 4.2.2

Apply the distributive property.

$\int \sec (x) \sec (x) + \sec (x) \tan (x) + \tan (x) (\sec (x) + \tan (x)) d x$

Step 4.2.3

Apply the distributive property.

$\int \sec (x) \sec (x) + \sec (x) \tan (x) + \tan (x) \sec (x) + \tan (x) \tan (x) d x$

Step 4.3

Simplify and combine like terms.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Multiply $\sec (x) \sec (x)$ .

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Step 4.3.1.1.1

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

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

Step 4.3.1.1.2

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

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

Step 4.3.1.1.3

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

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

Step 4.3.1.1.4

Add $1$ and $1$ .

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

Step 4.3.1.2

Multiply $\tan (x) \tan (x)$ .

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Step 4.3.1.2.1

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

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

Step 4.3.1.2.2

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

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

Step 4.3.1.2.3

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

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

Step 4.3.1.2.4

Add $1$ and $1$ .

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

Step 4.3.2

Reorder the factors of $\tan (x) \sec (x)$ .

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

Step 4.3.3

Add $\sec (x) \tan (x)$ and $\sec (x) \tan (x)$ .

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

Since the derivative of $\tan (x)$ is $\sec^{2} (x)$ , the integral of $\sec^{2} (x)$ is $\tan (x)$ .

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

Step 7

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

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

Step 8

Since the derivative of $\sec (x)$ is $\sec (x) \tan (x)$ , the integral of $\sec (x) \tan (x)$ is $\sec (x)$ .

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

Step 9

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

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

Step 10

Split the single integral into multiple integrals.

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

Step 11

Apply the constant rule.

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

Step 12

Since the derivative of $\tan (x)$ is $\sec^{2} (x)$ , the integral of $\sec^{2} (x)$ is $\tan (x)$ .

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

Step 13

Simplify.

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Step 13.1

Add $\tan (x)$ and $\tan (x)$ .

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

Step 13.2

Simplify.

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

Step 14

The answer is the antiderivative of the function $f (x) = {(\sec (x) + \tan (x))}^{2}$ .

$F (x) =$ $2 \sec (x) - x + 2 \tan (x) + C$