Calculus Examples

$f (x) = x^{2} + 2 x$ , $0 < x < 6$

Step 1

Check if $f (x) = x^{2} + 2 x$ is continuous.

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Step 1.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 1.2

$f (x)$ is continuous on $[0, 6]$ .

The function is continuous.

Step 2

Check if $f (x) = x^{2} + 2 x$ is differentiable.

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Step 2.1

Find the derivative.

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Step 2.1.1

Find the first derivative.

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Step 2.1.1.1

Differentiate.

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Step 2.1.1.1.1

By the Sum Rule, the derivative of $x^{2} + 2 x$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x]$

Step 2.1.1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x + \frac{d}{d x} [2 x]$

Step 2.1.1.2

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 2.1.1.2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 x + 2 \frac{d}{d x} [x]$

Step 2.1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 x + 2 \cdot 1$

Step 2.1.1.2.3

Multiply $2$ by $1$ .

$f' (x) = 2 x + 2$

Step 2.1.2

The first derivative of $f (x)$ with respect to $x$ is $2 x + 2$ .

$2 x + 2$

Step 2.2

Find if the derivative is continuous on $[0, 6]$ .

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Step 2.2.1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 2.2.2

$f' (x)$ is continuous on $[0, 6]$ .

The function is continuous.

Step 2.3

The function is differentiable on $[0, 6]$ because the derivative is continuous on $[0, 6]$ .

The function is differentiable.

Step 3

For arc length to be guaranteed, the function and its derivative must both be continuous on the closed interval $[0, 6]$ .

The function and its derivative are continuous on the closed interval $[0, 6]$ .

Step 4

Find the derivative of $f (x) = x^{2} + 2 x$ .

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Step 4.1

Differentiate.

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Step 4.1.1

By the Sum Rule, the derivative of $x^{2} + 2 x$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x]$

Step 4.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x + \frac{d}{d x} [2 x]$

Step 4.2

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 4.2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 x + 2 \frac{d}{d x} [x]$

Step 4.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 x + 2 \cdot 1$

Step 4.2.3

Multiply $2$ by $1$ .

$2 x + 2$

Step 5

To find the arc length of a function, use the formula $L = \int_{a}^{b} \sqrt{1 + {(f' (x))}^{2}} d x$ .

$\int_{0}^{6} \sqrt{1 + {(2 x + 2)}^{2}} d x$

Step 6

Evaluate the integral.

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Step 6.1

Complete the square.

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Step 6.1.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 4$

$b = 8$

$c = 5$

Step 6.1.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 6.1.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 6.1.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{8}{2 \cdot 4}$

Step 6.1.3.2

Simplify the right side.

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Step 6.1.3.2.1

Cancel the common factor of $8$ and $2$ .

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Step 6.1.3.2.1.1

Factor $2$ out of $8$ .

$d = \frac{2 \cdot 4}{2 \cdot 4}$

Step 6.1.3.2.1.2

Cancel the common factors.

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Step 6.1.3.2.1.2.1

Factor $2$ out of $2 \cdot 4$ .

$d = \frac{2 \cdot 4}{2 (4)}$

Step 6.1.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 4}{2 \cdot 4}$

Step 6.1.3.2.1.2.3

Rewrite the expression.

$d = \frac{4}{4}$

Step 6.1.3.2.2

Cancel the common factor of $4$ .

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Step 6.1.3.2.2.1

Cancel the common factor.

$d = \frac{4}{4}$

Step 6.1.3.2.2.2

Rewrite the expression.

$d = 1$

Step 6.1.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 6.1.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 5 - \frac{8^{2}}{4 \cdot 4}$

Step 6.1.4.2

Simplify the right side.

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Step 6.1.4.2.1

Simplify each term.

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Step 6.1.4.2.1.1

Raise $8$ to the power of $2$ .

$e = 5 - \frac{64}{4 \cdot 4}$

Step 6.1.4.2.1.2

Multiply $4$ by $4$ .

$e = 5 - \frac{64}{16}$

Step 6.1.4.2.1.3

Divide $64$ by $16$ .

$e = 5 - 1 \cdot 4$

Step 6.1.4.2.1.4

Multiply $- 1$ by $4$ .

$e = 5 - 4$

Step 6.1.4.2.2

Subtract $4$ from $5$ .

$e = 1$

Step 6.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $4 {(x + 1)}^{2} + 1$ .

$\int_{0}^{6} \sqrt{4 {(x + 1)}^{2} + 1} d x$

Step 6.2

Let $u = x + 1$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.2.1

Let $u = x + 1$ . Find $\frac{d u}{d x}$ .

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Step 6.2.1.1

Differentiate $x + 1$ .

$\frac{d}{d x} [x + 1]$

Step 6.2.1.2

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 6.2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [1]$

Step 6.2.1.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$1 + 0$

Step 6.2.1.5

Add $1$ and $0$ .

$1$

Step 6.2.2

Substitute the lower limit in for $x$ in $u = x + 1$ .

$u_{lower} = 0 + 1$

Step 6.2.3

Add $0$ and $1$ .

$u_{lower} = 1$

Step 6.2.4

Substitute the upper limit in for $x$ in $u = x + 1$ .

$u_{upper} = 6 + 1$

Step 6.2.5

Add $6$ and $1$ .

$u_{upper} = 7$

Step 6.2.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 1$

$u_{upper} = 7$

Step 6.2.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{1}^{7} \sqrt{4 u^{2} + 1} d u$

Step 6.3

Let $u = \frac{1}{2} \tan (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d u = \frac{\sec^{2} (t)}{2} d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\frac{\sec^{2} (t)}{2}$ is positive.

$\int_{1.10714871}^{1.49948886} \sqrt{4 {(\frac{1}{2} \tan (t))}^{2} + 1} \frac{\sec^{2} (t)}{2} d t$

Step 6.4

Simplify terms.

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Step 6.4.1

Simplify $\sqrt{4 {(\frac{1}{2} \tan (t))}^{2} + 1}$ .

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Step 6.4.1.1

Simplify each term.

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Step 6.4.1.1.1

Combine $\frac{1}{2}$ and $\tan (t)$ .

$\int_{1.10714871}^{1.49948886} \sqrt{4 {(\frac{\tan (t)}{2})}^{2} + 1} \frac{\sec^{2} (t)}{2} d t$

Step 6.4.1.1.2

Apply the product rule to $\frac{\tan (t)}{2}$ .

$\int_{1.10714871}^{1.49948886} \sqrt{4 \frac{\tan^{2} (t)}{2^{2}} + 1} \frac{\sec^{2} (t)}{2} d t$

Step 6.4.1.1.3

Raise $2$ to the power of $2$ .

$\int_{1.10714871}^{1.49948886} \sqrt{4 \frac{\tan^{2} (t)}{4} + 1} \frac{\sec^{2} (t)}{2} d t$

Step 6.4.1.1.4

Cancel the common factor of $4$ .

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Step 6.4.1.1.4.1

Cancel the common factor.

$\int_{1.10714871}^{1.49948886} \sqrt{4 \frac{\tan^{2} (t)}{4} + 1} \frac{\sec^{2} (t)}{2} d t$

Step 6.4.1.1.4.2

Rewrite the expression.

$\int_{1.10714871}^{1.49948886} \sqrt{\tan^{2} (t) + 1} \frac{\sec^{2} (t)}{2} d t$

Step 6.4.1.2

Apply pythagorean identity.

$\int_{1.10714871}^{1.49948886} \sqrt{\sec^{2} (t)} \frac{\sec^{2} (t)}{2} d t$

Step 6.4.1.3

Pull terms out from under the radical, assuming positive real numbers.

$\int_{1.10714871}^{1.49948886} \sec (t) \frac{\sec^{2} (t)}{2} d t$

Step 6.4.2

Simplify.

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Step 6.4.2.1

Combine $\sec (t)$ and $\frac{\sec^{2} (t)}{2}$ .

$\int_{1.10714871}^{1.49948886} \frac{\sec (t) \sec^{2} (t)}{2} d t$

Step 6.4.2.2

Multiply $\sec (t)$ by $\sec^{2} (t)$ by adding the exponents.

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Step 6.4.2.2.1

Multiply $\sec (t)$ by $\sec^{2} (t)$ .

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Step 6.4.2.2.1.1

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

$\int_{1.10714871}^{1.49948886} \frac{\sec^{1} (t) \sec^{2} (t)}{2} d t$

Step 6.4.2.2.1.2

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

$\int_{1.10714871}^{1.49948886} \frac{{\sec (t)}^{1 + 2}}{2} d t$

Step 6.4.2.2.2

Add $1$ and $2$ .

$\int_{1.10714871}^{1.49948886} \frac{\sec^{3} (t)}{2} d t$

Step 6.5

Since $\frac{1}{2}$ is constant with respect to $t$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int_{1.10714871}^{1.49948886} \sec^{3} (t) d t$

Step 6.6

Apply the reduction formula.

$\frac{1}{2} (\frac{\tan (t) \sec (t)}{2}]_{1.10714871}^{1.49948886} + \frac{1}{2} \int_{1.10714871}^{1.49948886} \sec (t) d t)$

Step 6.7

The integral of $\sec (t)$ with respect to $t$ is $\ln (| \sec (t) + \tan (t) |)$ .

$\frac{1}{2} (\frac{\tan (t) \sec (t)}{2}]_{1.10714871}^{1.49948886} + \frac{1}{2} \ln (| \sec (t) + \tan (t) |)]_{1.10714871}^{1.49948886})$

Step 6.8

Simplify.

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Step 6.8.1

Combine $\frac{1}{2}$ and $\ln (| \sec (t) + \tan (t) |)]_{1.10714871}^{1.49948886}$ .

$\frac{1}{2} (\frac{\tan (t) \sec (t)}{2}]_{1.10714871}^{1.49948886} + \frac{\ln (| \sec (t) + \tan (t) |)]_{1.10714871}^{1.49948886}}{2})$

Step 6.8.2

To write $\frac{\tan (t) \sec (t)}{2}]_{1.10714871}^{1.49948886}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} (\frac{\tan (t) \sec (t)}{2}]_{1.10714871}^{1.49948886} \cdot \frac{2}{2} + \frac{\ln (| \sec (t) + \tan (t) |)]_{1.10714871}^{1.49948886}}{2})$

Step 6.8.3

Combine $\frac{\tan (t) \sec (t)}{2}]_{1.10714871}^{1.49948886}$ and $\frac{2}{2}$ .

$\frac{1}{2} (\frac{\frac{\tan (t) \sec (t)}{2}]_{1.10714871}^{1.49948886} \cdot 2}{2} + \frac{\ln (| \sec (t) + \tan (t) |)]_{1.10714871}^{1.49948886}}{2})$

Step 6.8.4

Combine the numerators over the common denominator.

$\frac{1}{2} \cdot \frac{\frac{\tan (t) \sec (t)}{2}]_{1.10714871}^{1.49948886} \cdot 2 + \ln (| \sec (t) + \tan (t) |)]_{1.10714871}^{1.49948886}}{2}$

Step 6.8.5

Move $2$ to the left of $\frac{\tan (t) \sec (t)}{2}]_{1.10714871}^{1.49948886}$ .

$\frac{1}{2} \cdot \frac{2 \cdot (\frac{\tan (t) \sec (t)}{2}]_{1.10714871}^{1.49948886}) + \ln (| \sec (t) + \tan (t) |)]_{1.10714871}^{1.49948886}}{2}$

Step 6.8.6

Multiply $\frac{1}{2}$ by $\frac{2 (\frac{\tan (t) \sec (t)}{2}]_{1.10714871}^{1.49948886}) + \ln (| \sec (t) + \tan (t) |)]_{1.10714871}^{1.49948886}}{2}$ .

$\frac{2 (\frac{\tan (t) \sec (t)}{2}]_{1.10714871}^{1.49948886}) + \ln (| \sec (t) + \tan (t) |)]_{1.10714871}^{1.49948886}}{2 \cdot 2}$

Step 6.8.7

Multiply $2$ by $2$ .

$\frac{2 (\frac{\tan (t) \sec (t)}{2}]_{1.10714871}^{1.49948886}) + \ln (| \sec (t) + \tan (t) |)]_{1.10714871}^{1.49948886}}{4}$

Step 6.9

Substitute and simplify.

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Step 6.9.1

Evaluate $\frac{\tan (t) \sec (t)}{2}$ at $1.49948886$ and at $1.10714871$ .

$\frac{2 ((\frac{\tan (1.49948886) \sec (1.49948886)}{2}) - \frac{\tan (1.10714871) \sec (1.10714871)}{2}) + \ln (| \sec (t) + \tan (t) |)]_{1.10714871}^{1.49948886}}{4}$

Step 6.9.2

Evaluate $\ln (| \sec (t) + \tan (t) |)$ at $1.49948886$ and at $1.10714871$ .

$\frac{2 ((\frac{\tan (1.49948886) \sec (1.49948886)}{2}) - \frac{\tan (1.10714871) \sec (1.10714871)}{2}) + (\ln (| \sec (1.49948886) + \tan (1.49948886) |)) - \ln (| \sec (1.10714871) + \tan (1.10714871) |)}{4}$

Step 6.9.3

Remove unnecessary parentheses.

$\frac{2 (\frac{\tan (1.49948886) \sec (1.49948886)}{2} - \frac{\tan (1.10714871) \sec (1.10714871)}{2}) + \ln (| \sec (1.49948886) + \tan (1.49948886) |) - \ln (| \sec (1.10714871) + \tan (1.10714871) |)}{4}$

Step 6.10

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\frac{2 (\frac{\tan (1.49948886) \sec (1.49948886)}{2} - \frac{\tan (1.10714871) \sec (1.10714871)}{2}) + \ln (\frac{| \sec (1.49948886) + \tan (1.49948886) |}{| \sec (1.10714871) + \tan (1.10714871) |})}{4}$

Step 6.11

Simplify.

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Step 6.11.1

Simplify the numerator.

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Step 6.11.1.1

Evaluate $\tan (1.10714871)$ .

$\frac{2 (\frac{\tan (1.49948886) \sec (1.49948886)}{2} - \frac{2 \sec (1.10714871)}{2}) + \ln (\frac{| \sec (1.49948886) + \tan (1.49948886) |}{| \sec (1.10714871) + \tan (1.10714871) |})}{4}$

Step 6.11.1.2

Evaluate $\sec (1.10714871)$ .

$\frac{2 (\frac{\tan (1.49948886) \sec (1.49948886)}{2} - \frac{2 \cdot 2.23606797}{2}) + \ln (\frac{| \sec (1.49948886) + \tan (1.49948886) |}{| \sec (1.10714871) + \tan (1.10714871) |})}{4}$

Step 6.11.2

Multiply $2$ by $2.23606797$ .

$\frac{2 (\frac{\tan (1.49948886) \sec (1.49948886)}{2} - \frac{4.47213595}{2}) + \ln (\frac{| \sec (1.49948886) + \tan (1.49948886) |}{| \sec (1.10714871) + \tan (1.10714871) |})}{4}$

Step 6.11.3

Divide $4.47213595$ by $2$ .

$\frac{2 (\frac{\tan (1.49948886) \sec (1.49948886)}{2} - 1 \cdot 2.23606797) + \ln (\frac{| \sec (1.49948886) + \tan (1.49948886) |}{| \sec (1.10714871) + \tan (1.10714871) |})}{4}$

Step 6.11.4

Multiply $- 1$ by $2.23606797$ .

$\frac{2 (\frac{\tan (1.49948886) \sec (1.49948886)}{2} - 2.23606797) + \ln (\frac{| \sec (1.49948886) + \tan (1.49948886) |}{| \sec (1.10714871) + \tan (1.10714871) |})}{4}$

Step 6.11.5

Simplify each term.

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Step 6.11.5.1

Simplify the numerator.

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Step 6.11.5.1.1

Evaluate $\tan (1.49948886)$ .

$\frac{2 (\frac{14 \sec (1.49948886)}{2} - 2.23606797) + \ln (\frac{| \sec (1.49948886) + \tan (1.49948886) |}{| \sec (1.10714871) + \tan (1.10714871) |})}{4}$

Step 6.11.5.1.2

Evaluate $\sec (1.49948886)$ .

$\frac{2 (\frac{14 \cdot 14.03566884}{2} - 2.23606797) + \ln (\frac{| \sec (1.49948886) + \tan (1.49948886) |}{| \sec (1.10714871) + \tan (1.10714871) |})}{4}$

Step 6.11.5.2

Multiply $14$ by $14.03566884$ .

$\frac{2 (\frac{196.49936386}{2} - 2.23606797) + \ln (\frac{| \sec (1.49948886) + \tan (1.49948886) |}{| \sec (1.10714871) + \tan (1.10714871) |})}{4}$

Step 6.11.5.3

Divide $196.49936386$ by $2$ .

$\frac{2 (98.24968193 - 2.23606797) + \ln (\frac{| \sec (1.49948886) + \tan (1.49948886) |}{| \sec (1.10714871) + \tan (1.10714871) |})}{4}$

Step 6.11.6

Subtract $2.23606797$ from $98.24968193$ .

$\frac{2 \cdot 96.01361395 + \ln (\frac{| \sec (1.49948886) + \tan (1.49948886) |}{| \sec (1.10714871) + \tan (1.10714871) |})}{4}$

Step 6.11.7

Multiply $2$ by $96.01361395$ .

$\frac{192.02722791 + \ln (\frac{| \sec (1.49948886) + \tan (1.49948886) |}{| \sec (1.10714871) + \tan (1.10714871) |})}{4}$

Step 6.11.8

$\sec (1.49948886) + \tan (1.49948886)$ is approximately $28.03566884$ which is positive so remove the absolute value

$\frac{192.02722791 + \ln (\frac{\sec (1.49948886) + \tan (1.49948886)}{| \sec (1.10714871) + \tan (1.10714871) |})}{4}$

Step 6.11.9

$\sec (1.10714871) + \tan (1.10714871)$ is approximately $4.23606797$ which is positive so remove the absolute value

$\frac{192.02722791 + \ln (\frac{\sec (1.49948886) + \tan (1.49948886)}{\sec (1.10714871) + \tan (1.10714871)})}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{192.02722791 + \ln (\frac{\sec (1.49948886) + \tan (1.49948886)}{\sec (1.10714871) + \tan (1.10714871)})}{4}$

Decimal Form:

$48.47926750 \dots$

Step 8