Calculus Examples

2 ln (sec (x))

$2 \ln (\sec (x))$

Step 1

Find the asymptotes.

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Step 1.1

For any

$y = \sec (x)$ , vertical asymptotes occur at

$x = \frac{π}{2} + n π$ , where

$n$ is an integer. Use the basic period for

$y = s e c (x)$ ,

$(- \frac{π}{2}, \frac{3 π}{2})$ , to find the vertical asymptotes for

$y = \ln (\sec^{2} (x))$ . Set the inside of the secant function,

$b x + c$ , for

$y = a \sec (b x + c) + d$ equal to

$- \frac{π}{2}$ to find where the vertical asymptote occurs for

$y = \ln (\sec^{2} (x))$ .

$\sec^{2} (x) = - \frac{π}{2}$

Step 1.2

Solve for

$x$ .

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Step 1.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\sec (x) = \pm \sqrt{- \frac{π}{2}}$

Step 1.2.2

Simplify

$\pm \sqrt{- \frac{π}{2}}$ .

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Step 1.2.2.1

Rewrite

$- 1$ as

$i^{2}$ .

$\sec (x) = \pm \sqrt{i^{2} \frac{π}{2}}$

Step 1.2.2.2

Pull terms out from under the radical.

$\sec (x) = \pm i \sqrt{\frac{π}{2}}$

Step 1.2.2.3

Rewrite

$\sqrt{\frac{π}{2}}$ as

$\frac{\sqrt{π}}{\sqrt{2}}$ .

$\sec (x) = \pm i \frac{\sqrt{π}}{\sqrt{2}}$

Step 1.2.2.4

Multiply

$\frac{\sqrt{π}}{\sqrt{2}}$ by

$\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sec (x) = \pm i (\frac{\sqrt{π}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 1.2.2.5

Combine and simplify the denominator.

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Step 1.2.2.5.1

Multiply

$\frac{\sqrt{π}}{\sqrt{2}}$ by

$\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sec (x) = \pm i \frac{\sqrt{π} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 1.2.2.5.2

Raise

$\sqrt{2}$ to the power of

$1$ .

$\sec (x) = \pm i \frac{\sqrt{π} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 1.2.2.5.3

Raise

$\sqrt{2}$ to the power of

$1$ .

$\sec (x) = \pm i \frac{\sqrt{π} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 1.2.2.5.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sec (x) = \pm i \frac{\sqrt{π} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 1.2.2.5.5

Add

$1$ and

$1$ .

$\sec (x) = \pm i \frac{\sqrt{π} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 1.2.2.5.6

Rewrite

${\sqrt{2}}^{2}$ as

$2$ .

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Step 1.2.2.5.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2}$ as

$2^{\frac{1}{2}}$ .

$\sec (x) = \pm i \frac{\sqrt{π} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 1.2.2.5.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\sec (x) = \pm i \frac{\sqrt{π} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 1.2.2.5.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$\sec (x) = \pm i \frac{\sqrt{π} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.2.2.5.6.4

Cancel the common factor of

$2$ .

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Step 1.2.2.5.6.4.1

Cancel the common factor.

$\sec (x) = \pm i \frac{\sqrt{π} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.2.2.5.6.4.2

Rewrite the expression.

$\sec (x) = \pm i \frac{\sqrt{π} \sqrt{2}}{2^{1}}$

Step 1.2.2.5.6.5

Evaluate the exponent.

$\sec (x) = \pm i \frac{\sqrt{π} \sqrt{2}}{2}$

Step 1.2.2.6

Combine using the product rule for radicals.

$\sec (x) = \pm i \frac{\sqrt{π \cdot 2}}{2}$

Step 1.2.2.7

Combine

$i$ and

$\frac{\sqrt{π \cdot 2}}{2}$ .

$\sec (x) = \pm \frac{i \sqrt{π \cdot 2}}{2}$

Step 1.2.2.8

Move

$2$ to the left of

$π$ .

$\sec (x) = \pm \frac{i \sqrt{2 π}}{2}$

Step 1.2.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.3.1

First, use the positive value of the

$\pm$ to find the first solution.

$\sec (x) = \frac{i \sqrt{2 π}}{2}$

Step 1.2.3.2

Next, use the negative value of the

$\pm$ to find the second solution.

$\sec (x) = - \frac{i \sqrt{2 π}}{2}$

Step 1.2.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$\sec (x) = \frac{i \sqrt{2 π}}{2}, - \frac{i \sqrt{2 π}}{2}$

Step 1.2.4

Set up each of the solutions to solve for

$x$ .

$\sec (x) = \frac{i \sqrt{2 π}}{2}$

$\sec (x) = - \frac{i \sqrt{2 π}}{2}$

Step 1.2.5

Solve for

$x$ in

$\sec (x) = \frac{i \sqrt{2 π}}{2}$ .

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Step 1.2.5.1

Take the inverse secant of both sides of the equation to extract

$x$ from inside the secant.

$x = arcsec (\frac{i \sqrt{2 π}}{2})$

Step 1.2.5.2

The inverse secant of

$arcsec (\frac{i \sqrt{2 π}}{2})$ is undefined.

Undefined

Step 1.2.6

Solve for

$x$ in

$\sec (x) = - \frac{i \sqrt{2 π}}{2}$ .

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Step 1.2.6.1

Take the inverse secant of both sides of the equation to extract

$x$ from inside the secant.

$x = arcsec (- \frac{i \sqrt{2 π}}{2})$

Step 1.2.6.2

The inverse secant of

$arcsec (- \frac{i \sqrt{2 π}}{2})$ is undefined.

Undefined

Step 1.2.7

List all of the solutions.

No solution

Step 1.3

Set the inside of the secant function

$\sec^{2} (x)$ equal to

$\frac{3 π}{2}$ .

$\sec^{2} (x) = \frac{3 π}{2}$

Step 1.4

Solve for

$x$ .

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Step 1.4.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$\sec (x) = \pm \sqrt{\frac{3 π}{2}}$

Step 1.4.2

Simplify

$\pm \sqrt{\frac{3 π}{2}}$ .

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Step 1.4.2.1

Rewrite

$\sqrt{\frac{3 π}{2}}$ as

$\frac{\sqrt{3 π}}{\sqrt{2}}$ .

$\sec (x) = \pm \frac{\sqrt{3 π}}{\sqrt{2}}$

Step 1.4.2.2

Multiply

$\frac{\sqrt{3 π}}{\sqrt{2}}$ by

$\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sec (x) = \pm \frac{\sqrt{3 π}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 1.4.2.3

Combine and simplify the denominator.

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Step 1.4.2.3.1

Multiply

$\frac{\sqrt{3 π}}{\sqrt{2}}$ by

$\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sec (x) = \pm \frac{\sqrt{3 π} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 1.4.2.3.2

Raise

$\sqrt{2}$ to the power of

$1$ .

$\sec (x) = \pm \frac{\sqrt{3 π} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 1.4.2.3.3

Raise

$\sqrt{2}$ to the power of

$1$ .

$\sec (x) = \pm \frac{\sqrt{3 π} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 1.4.2.3.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sec (x) = \pm \frac{\sqrt{3 π} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 1.4.2.3.5

Add

$1$ and

$1$ .

$\sec (x) = \pm \frac{\sqrt{3 π} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 1.4.2.3.6

Rewrite

${\sqrt{2}}^{2}$ as

$2$ .

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Step 1.4.2.3.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2}$ as

$2^{\frac{1}{2}}$ .

$\sec (x) = \pm \frac{\sqrt{3 π} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 1.4.2.3.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\sec (x) = \pm \frac{\sqrt{3 π} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 1.4.2.3.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$\sec (x) = \pm \frac{\sqrt{3 π} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.4.2.3.6.4

Cancel the common factor of

$2$ .

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Step 1.4.2.3.6.4.1

Cancel the common factor.

$\sec (x) = \pm \frac{\sqrt{3 π} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 1.4.2.3.6.4.2

Rewrite the expression.

$\sec (x) = \pm \frac{\sqrt{3 π} \sqrt{2}}{2^{1}}$

Step 1.4.2.3.6.5

Evaluate the exponent.

$\sec (x) = \pm \frac{\sqrt{3 π} \sqrt{2}}{2}$

Step 1.4.2.4

Simplify the numerator.

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Step 1.4.2.4.1

Combine using the product rule for radicals.

$\sec (x) = \pm \frac{\sqrt{3 π \cdot 2}}{2}$

Step 1.4.2.4.2

Multiply

$2$ by

$3$ .

$\sec (x) = \pm \frac{\sqrt{6 π}}{2}$

Step 1.4.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.4.3.1

First, use the positive value of the

$\pm$ to find the first solution.

$\sec (x) = \frac{\sqrt{6 π}}{2}$

Step 1.4.3.2

Next, use the negative value of the

$\pm$ to find the second solution.

$\sec (x) = - \frac{\sqrt{6 π}}{2}$

Step 1.4.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$\sec (x) = \frac{\sqrt{6 π}}{2}, - \frac{\sqrt{6 π}}{2}$

Step 1.4.4

Set up each of the solutions to solve for

$x$ .

$\sec (x) = \frac{\sqrt{6 π}}{2}$

$\sec (x) = - \frac{\sqrt{6 π}}{2}$

Step 1.4.5

Solve for

$x$ in

$\sec (x) = \frac{\sqrt{6 π}}{2}$ .

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Step 1.4.5.1

Take the inverse secant of both sides of the equation to extract

$x$ from inside the secant.

$x = arcsec (\frac{\sqrt{6 π}}{2})$

Step 1.4.5.2

Simplify the right side.

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Step 1.4.5.2.1

Evaluate

$arcsec (\frac{\sqrt{6 π}}{2})$ .

$x = 1.09205895$

Step 1.4.5.3

The secant function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from

$2 π$ to find the solution in the fourth quadrant.

$x = 2 (3.14159265) - 1.09205895$

Step 1.4.5.4

Solve for

$x$ .

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Step 1.4.5.4.1

Remove parentheses.

$x = 2 (3.14159265) - 1.09205895$

Step 1.4.5.4.2

Simplify

$2 (3.14159265) - 1.09205895$ .

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Step 1.4.5.4.2.1

Multiply

$2$ by

$3.14159265$ .

$x = 6.2831853 - 1.09205895$

Step 1.4.5.4.2.2

Subtract

$1.09205895$ from

$6.2831853$ .

$x = 5.19112635$

Step 1.4.5.5

Find the period of

$\sec (x)$ .

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Step 1.4.5.5.1

The period of the function can be calculated using

$\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 1.4.5.5.2

Replace

$b$ with

$1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 1.4.5.5.3

The absolute value is the distance between a number and zero. The distance between

$0$ and

$1$ is

$1$ .

$\frac{2 π}{1}$

Step 1.4.5.5.4

Divide

$2 π$ by

$1$ .

$2 π$

Step 1.4.5.6

The period of the

$\sec (x)$ function is

$2 π$ so values will repeat every

$2 π$ radians in both directions.

$x = 1.09205895 + 2 π n, 5.19112635 + 2 π n$ , for any integer

$n$

$x = 1.09205895 + 2 π n, 5.19112635 + 2 π n$ , for any integer

$n$

Step 1.4.6

Solve for

$x$ in

$\sec (x) = - \frac{\sqrt{6 π}}{2}$ .

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Step 1.4.6.1

Take the inverse secant of both sides of the equation to extract

$x$ from inside the secant.

$x = arcsec (- \frac{\sqrt{6 π}}{2})$

Step 1.4.6.2

Simplify the right side.

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Step 1.4.6.2.1

Evaluate

$arcsec (- \frac{\sqrt{6 π}}{2})$ .

$x = 2.0495337$

Step 1.4.6.3

The secant function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from

$2 π$ to find the solution in the third quadrant.

$x = 2 (3.14159265) - 2.0495337$

Step 1.4.6.4

Solve for

$x$ .

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Step 1.4.6.4.1

Remove parentheses.

$x = 2 (3.14159265) - 2.0495337$

Step 1.4.6.4.2

Simplify

$2 (3.14159265) - 2.0495337$ .

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Step 1.4.6.4.2.1

Multiply

$2$ by

$3.14159265$ .

$x = 6.2831853 - 2.0495337$

Step 1.4.6.4.2.2

Subtract

$2.0495337$ from

$6.2831853$ .

$x = 4.2336516$

Step 1.4.6.5

Find the period of

$\sec (x)$ .

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Step 1.4.6.5.1

The period of the function can be calculated using

$\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 1.4.6.5.2

Replace

$b$ with

$1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 1.4.6.5.3

The absolute value is the distance between a number and zero. The distance between

$0$ and

$1$ is

$1$ .

$\frac{2 π}{1}$

Step 1.4.6.5.4

Divide

$2 π$ by

$1$ .

$2 π$

Step 1.4.6.6

The period of the

$\sec (x)$ function is

$2 π$ so values will repeat every

$2 π$ radians in both directions.

$x = 2.0495337 + 2 π n, 4.2336516 + 2 π n$ , for any integer

$n$

$x = 2.0495337 + 2 π n, 4.2336516 + 2 π n$ , for any integer

$n$

Step 1.4.7

List all of the solutions.

$x = 1.09205895 + 2 π n, 5.19112635 + 2 π n, 2.0495337 + 2 π n, 4.2336516 + 2 π n$ , for any integer

$n$

Step 1.4.8

Consolidate the solutions.

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Step 1.4.8.1

Consolidate

$1.09205895 + 2 π n$ and

$4.2336516 + 2 π n$ to

$1.09205895 + π n$ .

$x = 1.09205895 + π n, 5.19112635 + 2 π n, 2.0495337 + 2 π n$ , for any integer

$n$

Step 1.4.8.2

Consolidate

$5.19112635 + 2 π n$ and

$2.0495337 + 2 π n$ to

$2.0495337 + π n$ .

$x = 1.09205895 + π n, 2.0495337 + π n$ , for any integer

$n$

$x = 1.09205895 + π n, 2.0495337 + π n$ , for any integer

$n$

$x = 1.09205895 + π n, 2.0495337 + π n$ , for any integer

$n$

Step 1.5

The basic period for

$y = \ln (\sec^{2} (x))$ will occur at

$(, 1.09205895 + π n, 2.0495337 + π n)$ , where

and

$1.09205895 + π n, 2.0495337 + π n$ are vertical asymptotes.

$(, 1.09205895 + π n, 2.0495337 + π n)$

Step 1.6

Find the period

$\frac{2 π}{| b |}$ to find where the vertical asymptotes exist. Vertical asymptotes occur every half period.

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Step 1.6.1

The absolute value is the distance between a number and zero. The distance between

$0$ and

$1$ is

$1$ .

$\frac{2 π}{1}$

Step 1.6.2

Divide

$2 π$ by

$1$ .

$2 π$

Step 1.7

The vertical asymptotes for

$y = \ln (\sec^{2} (x))$ occur at

$1.09205895 + π n, 2.0495337 + π n$ , and every

$π n$ , where

$n$ is an integer. This is half of the period.

$π n$

Step 1.8

There are only vertical asymptotes for secant and cosecant functions.

Vertical Asymptotes:

$x = π n$ for any integer

$n$

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes:

$x = π n$ for any integer

$n$

No Horizontal Asymptotes

No Oblique Asymptotes

Step 2

Find the point at

$x = 1$ .

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Step 2.1

Replace the variable

$x$ with

$1$ in the expression.

$f (1) = 2 \ln (\sec (1))$

Step 2.2

Simplify the result.

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Step 2.2.1

Evaluate

$\sec (1)$ .

$f (1) = 2 \ln (1.85081571)$

Step 2.2.2

Simplify

$2 \ln (1.85081571)$ by moving

$2$ inside the logarithm.

$f (1) = \ln ({1.85081571}^{2})$

Step 2.2.3

Raise

$1.85081571$ to the power of

$2$ .

$f (1) = \ln (3.42551882)$

Step 2.2.4

The final answer is

$\ln (3.42551882)$ .

$\ln (3.42551882)$

Step 3

Find the point at

$x = 5$ .

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Step 3.1

Replace the variable

$x$ with

$5$ in the expression.

$f (5) = 2 \ln (\sec (5))$

Step 3.2

Simplify the result.

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Step 3.2.1

Evaluate

$\sec (5)$ .

$f (5) = 2 \ln (3.52532008)$

Step 3.2.2

Simplify

$2 \ln (3.52532008)$ by moving

$2$ inside the logarithm.

$f (5) = \ln ({3.52532008}^{2})$

Step 3.2.3

Raise

$3.52532008$ to the power of

$2$ .

$f (5) = \ln (12.4278817)$

Step 3.2.4

The final answer is

$\ln (12.4278817)$ .

$\ln (12.4278817)$

Step 4

Find the point at

$x = 6$ .

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Step 4.1

Replace the variable

$x$ with

$6$ in the expression.

$f (6) = 2 \ln (\sec (6))$

Step 4.2

Simplify the result.

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Step 4.2.1

Evaluate

$\sec (6)$ .

$f (6) = 2 \ln (1.04148192)$

Step 4.2.2

Simplify

$2 \ln (1.04148192)$ by moving

$2$ inside the logarithm.

$f (6) = \ln ({1.04148192}^{2})$

Step 4.2.3

Raise

$1.04148192$ to the power of

$2$ .

$f (6) = \ln (1.0846846)$

Step 4.2.4

The final answer is

$\ln (1.0846846)$ .

$\ln (1.0846846)$

Step 5

The log function can be graphed using the vertical asymptote at

$x = π n (f o r) (a n y) (i n t e g e r) n$ and the points

$(1, 1.23125294), (5, 2.51994247), (6, 0.08128925)$ .

Vertical Asymptote:

$x = π n (f o r) (a n y) (i n t e g e r) n$

$\begin{array}{c} x & y \\ 1 & 1.231 \\ 5 & 2.52 \\ 6 & 0.081 \end{array}$

Step 6