Calculus Examples

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

Step 1

Find the asymptotes.

Tap for more steps...

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 $x$ .

Tap for more steps...

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 $\pm \sqrt{- \frac{π}{2}}$ .

Tap for more steps...

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.

Tap for more steps...

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 $2$ .

Tap for more steps...

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 $2$ .

Tap for more steps...

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.

Tap for more steps...

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

Tap for more steps...

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

Tap for more steps...

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 $x$ .

Tap for more steps...

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 $\pm \sqrt{\frac{3 π}{2}}$ .

Tap for more steps...

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.

Tap for more steps...

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 $2$ .

Tap for more steps...

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 $2$ .

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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 $\sec (x) = \frac{\sqrt{6 π}}{2}$ .

Tap for more steps...

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.

Tap for more steps...

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 $x$ .

Tap for more steps...

Step 1.4.5.4.1

Remove parentheses.

$x = 2 (3.14159265) - 1.09205895$

Step 1.4.5.4.2

Simplify $2 (3.14159265) - 1.09205895$ .

Tap for more steps...

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 $\sec (x)$ .

Tap for more steps...

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 $n$

Step 1.4.6

Solve for $x$ in $\sec (x) = - \frac{\sqrt{6 π}}{2}$ .

Tap for more steps...

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.

Tap for more steps...

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 $x$ .

Tap for more steps...

Step 1.4.6.4.1

Remove parentheses.

$x = 2 (3.14159265) - 2.0495337$

Step 1.4.6.4.2

Simplify $2 (3.14159265) - 2.0495337$ .

Tap for more steps...

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 $\sec (x)$ .

Tap for more steps...

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 $n$

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.

Tap for more steps...

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 $n$

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.

Tap for more steps...

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 $x = 1$ .

Tap for more steps...

Step 2.1

Replace the variable $x$ with $1$ in the expression.

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

Step 2.2

Simplify the result.

Tap for more steps...

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 $x = 5$ .

Tap for more steps...

Step 3.1

Replace the variable $x$ with $5$ in the expression.

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

Step 3.2

Simplify the result.

Tap for more steps...

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 $x = 6$ .

Tap for more steps...

Step 4.1

Replace the variable $x$ with $6$ in the expression.

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

Step 4.2

Simplify the result.

Tap for more steps...

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