Calculus Examples

$2 x^{2} + 3 x = 5$ , $(- 1, 5)$

Step 1

Subtract $5$ from both sides of the equation.

$2 x^{2} + 3 x - 5 = 0$

Step 2

Check if $f (x) = 2 x^{2} + 3 x - 5$ is continuous.

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Step 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

$f (x)$ is continuous on $[- 1, 5]$ .

The function is continuous.

Step 3

Check if $f (x) = 2 x^{2} + 3 x - 5$ is differentiable.

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Step 3.1

Find the derivative.

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Step 3.1.1

Find the first derivative.

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Step 3.1.1.1

By the Sum Rule, the derivative of $2 x^{2} + 3 x - 5$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 5]$ .

$\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 5]$

Step 3.1.1.2

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

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Step 3.1.1.2.1

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

$2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 5]$

Step 3.1.1.2.2

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

$2 (2 x) + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 5]$

Step 3.1.1.2.3

Multiply $2$ by $2$ .

$4 x + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 5]$

Step 3.1.1.3

Evaluate $\frac{d}{d x} [3 x]$ .

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Step 3.1.1.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$4 x + 3 \frac{d}{d x} [x] + \frac{d}{d x} [- 5]$

Step 3.1.1.3.2

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

$4 x + 3 \cdot 1 + \frac{d}{d x} [- 5]$

Step 3.1.1.3.3

Multiply $3$ by $1$ .

$4 x + 3 + \frac{d}{d x} [- 5]$

Step 3.1.1.4

Differentiate using the Constant Rule.

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Step 3.1.1.4.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$4 x + 3 + 0$

Step 3.1.1.4.2

Add $4 x + 3$ and $0$ .

$f' (x) = 4 x + 3$

Step 3.1.2

The first derivative of $f (x)$ with respect to $x$ is $4 x + 3$ .

$4 x + 3$

Step 3.2

Find if the derivative is continuous on $[- 1, 5]$ .

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Step 3.2.1

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 3.2.2

$f' (x)$ is continuous on $[- 1, 5]$ .

The function is continuous.

Step 3.3

The function is differentiable on $[- 1, 5]$ because the derivative is continuous on $[- 1, 5]$ .

The function is differentiable.

Step 4

For arc length to be guaranteed, the function and its derivative must both be continuous on the closed interval $[- 1, 5]$ .

The function and its derivative are continuous on the closed interval $[- 1, 5]$ .

Step 5

Find the derivative of $f (x) = 2 x^{2} + 3 x - 5$ .

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Step 5.1

By the Sum Rule, the derivative of $2 x^{2} + 3 x - 5$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 5]$ .

$\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 5]$

Step 5.2

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

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Step 5.2.1

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

$2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 5]$

Step 5.2.2

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

$2 (2 x) + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 5]$

Step 5.2.3

Multiply $2$ by $2$ .

$4 x + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 5]$

Step 5.3

Evaluate $\frac{d}{d x} [3 x]$ .

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Step 5.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$4 x + 3 \frac{d}{d x} [x] + \frac{d}{d x} [- 5]$

Step 5.3.2

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

$4 x + 3 \cdot 1 + \frac{d}{d x} [- 5]$

Step 5.3.3

Multiply $3$ by $1$ .

$4 x + 3 + \frac{d}{d x} [- 5]$

Step 5.4

Differentiate using the Constant Rule.

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Step 5.4.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$4 x + 3 + 0$

Step 5.4.2

Add $4 x + 3$ and $0$ .

$4 x + 3$

Step 6

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

$\int_{- 1}^{5} \sqrt{1 + {(4 x + 3)}^{2}} d x$

Step 7

Evaluate the integral.

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Step 7.1

Complete the square.

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Step 7.1.1

Simplify the expression.

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Step 7.1.1.1

Apply the distributive property.

$2 (8 x^{2}) + 2 (12 x) + 2 \cdot 5$

Step 7.1.1.2

Simplify.

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Step 7.1.1.2.1

Multiply $8$ by $2$ .

$16 x^{2} + 2 (12 x) + 2 \cdot 5$

Step 7.1.1.2.2

Multiply $12$ by $2$ .

$16 x^{2} + 24 x + 2 \cdot 5$

Step 7.1.1.2.3

Multiply $2$ by $5$ .

$16 x^{2} + 24 x + 10$

Step 7.1.2

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

$a = 16$

$b = 24$

$c = 10$

Step 7.1.3

Consider the vertex form of a parabola.

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

Step 7.1.4

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

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Step 7.1.4.1

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

$d = \frac{24}{2 \cdot 16}$

Step 7.1.4.2

Simplify the right side.

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Step 7.1.4.2.1

Cancel the common factor of $24$ and $2$ .

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Step 7.1.4.2.1.1

Factor $2$ out of $24$ .

$d = \frac{2 \cdot 12}{2 \cdot 16}$

Step 7.1.4.2.1.2

Cancel the common factors.

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Step 7.1.4.2.1.2.1

Factor $2$ out of $2 \cdot 16$ .

$d = \frac{2 \cdot 12}{2 (16)}$

Step 7.1.4.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 12}{2 \cdot 16}$

Step 7.1.4.2.1.2.3

Rewrite the expression.

$d = \frac{12}{16}$

Step 7.1.4.2.2

Cancel the common factor of $12$ and $16$ .

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Step 7.1.4.2.2.1

Factor $4$ out of $12$ .

$d = \frac{4 (3)}{16}$

Step 7.1.4.2.2.2

Cancel the common factors.

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Step 7.1.4.2.2.2.1

Factor $4$ out of $16$ .

$d = \frac{4 \cdot 3}{4 \cdot 4}$

Step 7.1.4.2.2.2.2

Cancel the common factor.

$d = \frac{4 \cdot 3}{4 \cdot 4}$

Step 7.1.4.2.2.2.3

Rewrite the expression.

$d = \frac{3}{4}$

Step 7.1.5

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

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Step 7.1.5.1

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

$e = 10 - \frac{24^{2}}{4 \cdot 16}$

Step 7.1.5.2

Simplify the right side.

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Step 7.1.5.2.1

Simplify each term.

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Step 7.1.5.2.1.1

Raise $24$ to the power of $2$ .

$e = 10 - \frac{576}{4 \cdot 16}$

Step 7.1.5.2.1.2

Multiply $4$ by $16$ .

$e = 10 - \frac{576}{64}$

Step 7.1.5.2.1.3

Divide $576$ by $64$ .

$e = 10 - 1 \cdot 9$

Step 7.1.5.2.1.4

Multiply $- 1$ by $9$ .

$e = 10 - 9$

Step 7.1.5.2.2

Subtract $9$ from $10$ .

$e = 1$

Step 7.1.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $16 {(x + \frac{3}{4})}^{2} + 1$ .

$\int_{- 1}^{5} \sqrt{16 {(x + \frac{3}{4})}^{2} + 1} d x$

Step 7.2

Let $u = x + \frac{3}{4}$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 7.2.1

Let $u = x + \frac{3}{4}$ . Find $\frac{d u}{d x}$ .

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Step 7.2.1.1

Differentiate $x + \frac{3}{4}$ .

$\frac{d}{d x} [x + \frac{3}{4}]$

Step 7.2.1.2

By the Sum Rule, the derivative of $x + \frac{3}{4}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [\frac{3}{4}]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [\frac{3}{4}]$

Step 7.2.1.3

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

$1 + \frac{d}{d x} [\frac{3}{4}]$

Step 7.2.1.4

Since $\frac{3}{4}$ is constant with respect to $x$ , the derivative of $\frac{3}{4}$ with respect to $x$ is $0$ .

$1 + 0$

Step 7.2.1.5

Add $1$ and $0$ .

$1$

Step 7.2.2

Substitute the lower limit in for $x$ in $u = x + \frac{3}{4}$ .

$u_{lower} = - 1 + \frac{3}{4}$

Step 7.2.3

Simplify.

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Step 7.2.3.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$u_{lower} = - 1 \cdot \frac{4}{4} + \frac{3}{4}$

Step 7.2.3.2

Combine $- 1$ and $\frac{4}{4}$ .

$u_{lower} = \frac{- 1 \cdot 4}{4} + \frac{3}{4}$

Step 7.2.3.3

Combine the numerators over the common denominator.

$u_{lower} = \frac{- 1 \cdot 4 + 3}{4}$

Step 7.2.3.4

Simplify the numerator.

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Step 7.2.3.4.1

Multiply $- 1$ by $4$ .

$u_{lower} = \frac{- 4 + 3}{4}$

Step 7.2.3.4.2

Add $- 4$ and $3$ .

$u_{lower} = \frac{- 1}{4}$

Step 7.2.3.5

Move the negative in front of the fraction.

$u_{lower} = - \frac{1}{4}$

Step 7.2.4

Substitute the upper limit in for $x$ in $u = x + \frac{3}{4}$ .

$u_{upper} = 5 + \frac{3}{4}$

Step 7.2.5

Simplify.

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Step 7.2.5.1

To write $5$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$u_{upper} = 5 \cdot \frac{4}{4} + \frac{3}{4}$

Step 7.2.5.2

Combine $5$ and $\frac{4}{4}$ .

$u_{upper} = \frac{5 \cdot 4}{4} + \frac{3}{4}$

Step 7.2.5.3

Combine the numerators over the common denominator.

$u_{upper} = \frac{5 \cdot 4 + 3}{4}$

Step 7.2.5.4

Simplify the numerator.

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Step 7.2.5.4.1

Multiply $5$ by $4$ .

$u_{upper} = \frac{20 + 3}{4}$

Step 7.2.5.4.2

Add $20$ and $3$ .

$u_{upper} = \frac{23}{4}$

Step 7.2.6

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

$u_{lower} = - \frac{1}{4}$

$u_{upper} = \frac{23}{4}$

Step 7.2.7

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

$\int_{- \frac{1}{4}}^{\frac{23}{4}} \sqrt{16 u^{2} + 1} d u$

Step 7.3

Let $u = \frac{1}{4} \tan (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d u = \frac{\sec^{2} (t)}{4} d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\frac{\sec^{2} (t)}{4}$ is positive.

$\int_{- \frac{π}{4}}^{1.52734543} \sqrt{16 {(\frac{1}{4} \tan (t))}^{2} + 1} \frac{\sec^{2} (t)}{4} d t$

Step 7.4

Simplify terms.

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Step 7.4.1

Simplify $\sqrt{16 {(\frac{1}{4} \tan (t))}^{2} + 1}$ .

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Step 7.4.1.1

Simplify each term.

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Step 7.4.1.1.1

Combine $\frac{1}{4}$ and $\tan (t)$ .

$\int_{- \frac{π}{4}}^{1.52734543} \sqrt{16 {(\frac{\tan (t)}{4})}^{2} + 1} \frac{\sec^{2} (t)}{4} d t$

Step 7.4.1.1.2

Apply the product rule to $\frac{\tan (t)}{4}$ .

$\int_{- \frac{π}{4}}^{1.52734543} \sqrt{16 \frac{\tan^{2} (t)}{4^{2}} + 1} \frac{\sec^{2} (t)}{4} d t$

Step 7.4.1.1.3

Raise $4$ to the power of $2$ .

$\int_{- \frac{π}{4}}^{1.52734543} \sqrt{16 \frac{\tan^{2} (t)}{16} + 1} \frac{\sec^{2} (t)}{4} d t$

Step 7.4.1.1.4

Cancel the common factor of $16$ .

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Step 7.4.1.1.4.1

Cancel the common factor.

$\int_{- \frac{π}{4}}^{1.52734543} \sqrt{16 \frac{\tan^{2} (t)}{16} + 1} \frac{\sec^{2} (t)}{4} d t$

Step 7.4.1.1.4.2

Rewrite the expression.

$\int_{- \frac{π}{4}}^{1.52734543} \sqrt{\tan^{2} (t) + 1} \frac{\sec^{2} (t)}{4} d t$

Step 7.4.1.2

Apply pythagorean identity.

$\int_{- \frac{π}{4}}^{1.52734543} \sqrt{\sec^{2} (t)} \frac{\sec^{2} (t)}{4} d t$

Step 7.4.1.3

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

$\int_{- \frac{π}{4}}^{1.52734543} \sec (t) \frac{\sec^{2} (t)}{4} d t$

Step 7.4.2

Simplify.

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Step 7.4.2.1

Combine $\sec (t)$ and $\frac{\sec^{2} (t)}{4}$ .

$\int_{- \frac{π}{4}}^{1.52734543} \frac{\sec (t) \sec^{2} (t)}{4} d t$

Step 7.4.2.2

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

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Step 7.4.2.2.1

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

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Step 7.4.2.2.1.1

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

$\int_{- \frac{π}{4}}^{1.52734543} \frac{\sec^{1} (t) \sec^{2} (t)}{4} d t$

Step 7.4.2.2.1.2

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

$\int_{- \frac{π}{4}}^{1.52734543} \frac{{\sec (t)}^{1 + 2}}{4} d t$

Step 7.4.2.2.2

Add $1$ and $2$ .

$\int_{- \frac{π}{4}}^{1.52734543} \frac{\sec^{3} (t)}{4} d t$

Step 7.5

Since $\frac{1}{4}$ is constant with respect to $t$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int_{- \frac{π}{4}}^{1.52734543} \sec^{3} (t) d t$

Step 7.6

Apply the reduction formula.

$\frac{1}{4} (\frac{\tan (t) \sec (t)}{2}]_{- \frac{π}{4}}^{1.52734543} + \frac{1}{2} \int_{- \frac{π}{4}}^{1.52734543} \sec (t) d t)$

Step 7.7

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

$\frac{1}{4} (\frac{\tan (t) \sec (t)}{2}]_{- \frac{π}{4}}^{1.52734543} + \frac{1}{2} \ln (| \sec (t) + \tan (t) |)]_{- \frac{π}{4}}^{1.52734543})$

Step 7.8

Simplify.

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Step 7.8.1

Combine $\frac{1}{2}$ and $\ln (| \sec (t) + \tan (t) |)]_{- \frac{π}{4}}^{1.52734543}$ .

$\frac{1}{4} (\frac{\tan (t) \sec (t)}{2}]_{- \frac{π}{4}}^{1.52734543} + \frac{\ln (| \sec (t) + \tan (t) |)]_{- \frac{π}{4}}^{1.52734543}}{2})$

Step 7.8.2

To write $\frac{\tan (t) \sec (t)}{2}]_{- \frac{π}{4}}^{1.52734543}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{4} (\frac{\tan (t) \sec (t)}{2}]_{- \frac{π}{4}}^{1.52734543} \cdot \frac{2}{2} + \frac{\ln (| \sec (t) + \tan (t) |)]_{- \frac{π}{4}}^{1.52734543}}{2})$

Step 7.8.3

Combine $\frac{\tan (t) \sec (t)}{2}]_{- \frac{π}{4}}^{1.52734543}$ and $\frac{2}{2}$ .

$\frac{1}{4} (\frac{\frac{\tan (t) \sec (t)}{2}]_{- \frac{π}{4}}^{1.52734543} \cdot 2}{2} + \frac{\ln (| \sec (t) + \tan (t) |)]_{- \frac{π}{4}}^{1.52734543}}{2})$

Step 7.8.4

Combine the numerators over the common denominator.

$\frac{1}{4} \cdot \frac{\frac{\tan (t) \sec (t)}{2}]_{- \frac{π}{4}}^{1.52734543} \cdot 2 + \ln (| \sec (t) + \tan (t) |)]_{- \frac{π}{4}}^{1.52734543}}{2}$

Step 7.8.5

Move $2$ to the left of $\frac{\tan (t) \sec (t)}{2}]_{- \frac{π}{4}}^{1.52734543}$ .

$\frac{1}{4} \cdot \frac{2 \cdot (\frac{\tan (t) \sec (t)}{2}]_{- \frac{π}{4}}^{1.52734543}) + \ln (| \sec (t) + \tan (t) |)]_{- \frac{π}{4}}^{1.52734543}}{2}$

Step 7.8.6

Multiply $\frac{1}{4}$ by $\frac{2 (\frac{\tan (t) \sec (t)}{2}]_{- \frac{π}{4}}^{1.52734543}) + \ln (| \sec (t) + \tan (t) |)]_{- \frac{π}{4}}^{1.52734543}}{2}$ .

$\frac{2 (\frac{\tan (t) \sec (t)}{2}]_{- \frac{π}{4}}^{1.52734543}) + \ln (| \sec (t) + \tan (t) |)]_{- \frac{π}{4}}^{1.52734543}}{4 \cdot 2}$

Step 7.8.7

Multiply $4$ by $2$ .

$\frac{2 (\frac{\tan (t) \sec (t)}{2}]_{- \frac{π}{4}}^{1.52734543}) + \ln (| \sec (t) + \tan (t) |)]_{- \frac{π}{4}}^{1.52734543}}{8}$

Step 7.9

Substitute and simplify.

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Step 7.9.1

Evaluate $\frac{\tan (t) \sec (t)}{2}$ at $1.52734543$ and at $- \frac{π}{4}$ .

$\frac{2 ((\frac{\tan (1.52734543) \sec (1.52734543)}{2}) - \frac{\tan (- \frac{π}{4}) \sec (- \frac{π}{4})}{2}) + \ln (| \sec (t) + \tan (t) |)]_{- \frac{π}{4}}^{1.52734543}}{8}$

Step 7.9.2

Evaluate $\ln (| \sec (t) + \tan (t) |)$ at $1.52734543$ and at $- \frac{π}{4}$ .

$\frac{2 ((\frac{\tan (1.52734543) \sec (1.52734543)}{2}) - \frac{\tan (- \frac{π}{4}) \sec (- \frac{π}{4})}{2}) + (\ln (| \sec (1.52734543) + \tan (1.52734543) |)) - \ln (| \sec (- \frac{π}{4}) + \tan (- \frac{π}{4}) |)}{8}$

Step 7.9.3

Remove unnecessary parentheses.

$\frac{2 (\frac{\tan (1.52734543) \sec (1.52734543)}{2} - \frac{\tan (- \frac{π}{4}) \sec (- \frac{π}{4})}{2}) + \ln (| \sec (1.52734543) + \tan (1.52734543) |) - \ln (| \sec (- \frac{π}{4}) + \tan (- \frac{π}{4}) |)}{8}$

Step 7.10

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

$\frac{2 (\frac{\tan (1.52734543) \sec (1.52734543)}{2} - \frac{\tan (- \frac{π}{4}) \sec (- \frac{π}{4})}{2}) + \ln (\frac{| \sec (1.52734543) + \tan (1.52734543) |}{| \sec (- \frac{π}{4}) + \tan (- \frac{π}{4}) |})}{8}$

Step 7.11

Simplify.

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Step 7.11.1

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{2 (\frac{\tan (1.52734543) \sec (1.52734543)}{2} - \frac{\tan (\frac{7 π}{4}) \sec (- \frac{π}{4})}{2}) + \ln (\frac{| \sec (1.52734543) + \tan (1.52734543) |}{| \sec (- \frac{π}{4}) + \tan (- \frac{π}{4}) |})}{8}$

Step 7.11.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because tangent is negative in the fourth quadrant.

$\frac{2 (\frac{\tan (1.52734543) \sec (1.52734543)}{2} - \frac{- \tan (\frac{π}{4}) \sec (- \frac{π}{4})}{2}) + \ln (\frac{| \sec (1.52734543) + \tan (1.52734543) |}{| \sec (- \frac{π}{4}) + \tan (- \frac{π}{4}) |})}{8}$

Step 7.11.3

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$\frac{2 (\frac{\tan (1.52734543) \sec (1.52734543)}{2} - \frac{- 1 \cdot 1 \sec (- \frac{π}{4})}{2}) + \ln (\frac{| \sec (1.52734543) + \tan (1.52734543) |}{| \sec (- \frac{π}{4}) + \tan (- \frac{π}{4}) |})}{8}$

Step 7.11.4

Multiply $- 1$ by $1$ .

$\frac{2 (\frac{\tan (1.52734543) \sec (1.52734543)}{2} - \frac{- \sec (- \frac{π}{4})}{2}) + \ln (\frac{| \sec (1.52734543) + \tan (1.52734543) |}{| \sec (- \frac{π}{4}) + \tan (- \frac{π}{4}) |})}{8}$

Step 7.11.5

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{2 (\frac{\tan (1.52734543) \sec (1.52734543)}{2} - \frac{- \sec (\frac{7 π}{4})}{2}) + \ln (\frac{| \sec (1.52734543) + \tan (1.52734543) |}{| \sec (- \frac{π}{4}) + \tan (- \frac{π}{4}) |})}{8}$

Step 7.11.6

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\frac{2 (\frac{\tan (1.52734543) \sec (1.52734543)}{2} - \frac{- \sec (\frac{π}{4})}{2}) + \ln (\frac{| \sec (1.52734543) + \tan (1.52734543) |}{| \sec (- \frac{π}{4}) + \tan (- \frac{π}{4}) |})}{8}$

Step 7.11.7

The exact value of $\sec (\frac{π}{4})$ is $\frac{2}{\sqrt{2}}$ .

$\frac{2 (\frac{\tan (1.52734543) \sec (1.52734543)}{2} - \frac{- \frac{2}{\sqrt{2}}}{2}) + \ln (\frac{| \sec (1.52734543) + \tan (1.52734543) |}{| \sec (- \frac{π}{4}) + \tan (- \frac{π}{4}) |})}{8}$

Step 7.11.8

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 (\frac{\tan (1.52734543) \sec (1.52734543)}{2} - (- \frac{2}{\sqrt{2}} \cdot \frac{1}{2})) + \ln (\frac{| \sec (1.52734543) + \tan (1.52734543) |}{| \sec (- \frac{π}{4}) + \tan (- \frac{π}{4}) |})}{8}$

Step 7.11.9

Cancel the common factor of $2$ .

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Step 7.11.9.1

Move the leading negative in $- \frac{2}{\sqrt{2}}$ into the numerator.

$\frac{2 (\frac{\tan (1.52734543) \sec (1.52734543)}{2} - (\frac{- 2}{\sqrt{2}} \cdot \frac{1}{2})) + \ln (\frac{| \sec (1.52734543) + \tan (1.52734543) |}{| \sec (- \frac{π}{4}) + \tan (- \frac{π}{4}) |})}{8}$

Step 7.11.9.2

Factor $2$ out of $- 2$ .

$\frac{2 (\frac{\tan (1.52734543) \sec (1.52734543)}{2} - (\frac{2 (- 1)}{\sqrt{2}} \cdot \frac{1}{2})) + \ln (\frac{| \sec (1.52734543) + \tan (1.52734543) |}{| \sec (- \frac{π}{4}) + \tan (- \frac{π}{4}) |})}{8}$

Step 7.11.9.3

Cancel the common factor.

$\frac{2 (\frac{\tan (1.52734543) \sec (1.52734543)}{2} - (\frac{2 \cdot - 1}{\sqrt{2}} \cdot \frac{1}{2})) + \ln (\frac{| \sec (1.52734543) + \tan (1.52734543) |}{| \sec (- \frac{π}{4}) + \tan (- \frac{π}{4}) |})}{8}$

Step 7.11.9.4

Rewrite the expression.

$\frac{2 (\frac{\tan (1.52734543) \sec (1.52734543)}{2} - \frac{- 1}{\sqrt{2}}) + \ln (\frac{| \sec (1.52734543) + \tan (1.52734543) |}{| \sec (- \frac{π}{4}) + \tan (- \frac{π}{4}) |})}{8}$

Step 7.11.10

Move the negative in front of the fraction.

$\frac{2 (\frac{\tan (1.52734543) \sec (1.52734543)}{2} - - \frac{1}{\sqrt{2}}) + \ln (\frac{| \sec (1.52734543) + \tan (1.52734543) |}{| \sec (- \frac{π}{4}) + \tan (- \frac{π}{4}) |})}{8}$

Step 7.11.11

Multiply $- - \frac{1}{\sqrt{2}}$ .

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Step 7.11.11.1

Multiply $- 1$ by $- 1$ .

$\frac{2 (\frac{\tan (1.52734543) \sec (1.52734543)}{2} + 1 \frac{1}{\sqrt{2}}) + \ln (\frac{| \sec (1.52734543) + \tan (1.52734543) |}{| \sec (- \frac{π}{4}) + \tan (- \frac{π}{4}) |})}{8}$

Step 7.11.11.2

Multiply $\frac{1}{\sqrt{2}}$ by $1$ .

$\frac{2 (\frac{\tan (1.52734543) \sec (1.52734543)}{2} + \frac{1}{\sqrt{2}}) + \ln (\frac{| \sec (1.52734543) + \tan (1.52734543) |}{| \sec (- \frac{π}{4}) + \tan (- \frac{π}{4}) |})}{8}$

Step 7.11.12

Simplify each term.

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Step 7.11.12.1

Simplify the numerator.

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Step 7.11.12.1.1

Evaluate $\tan (1.52734543)$ .

$\frac{2 (\frac{23 \sec (1.52734543)}{2} + \frac{1}{\sqrt{2}}) + \ln (\frac{| \sec (1.52734543) + \tan (1.52734543) |}{| \sec (- \frac{π}{4}) + \tan (- \frac{π}{4}) |})}{8}$

Step 7.11.12.1.2

Evaluate $\sec (1.52734543)$ .

$\frac{2 (\frac{23 \cdot 23.02172886}{2} + \frac{1}{\sqrt{2}}) + \ln (\frac{| \sec (1.52734543) + \tan (1.52734543) |}{| \sec (- \frac{π}{4}) + \tan (- \frac{π}{4}) |})}{8}$

Step 7.11.12.2

Multiply $23$ by $23.02172886$ .

$\frac{2 (\frac{529.49976392}{2} + \frac{1}{\sqrt{2}}) + \ln (\frac{| \sec (1.52734543) + \tan (1.52734543) |}{| \sec (- \frac{π}{4}) + \tan (- \frac{π}{4}) |})}{8}$

Step 7.11.12.3

Divide $529.49976392$ by $2$ .

$\frac{2 (264.74988196 + \frac{1}{\sqrt{2}}) + \ln (\frac{| \sec (1.52734543) + \tan (1.52734543) |}{| \sec (- \frac{π}{4}) + \tan (- \frac{π}{4}) |})}{8}$

Step 7.11.13

Add $264.74988196$ and $\frac{1}{\sqrt{2}}$ .

$\frac{2 \cdot 265.45698874 + \ln (\frac{| \sec (1.52734543) + \tan (1.52734543) |}{| \sec (- \frac{π}{4}) + \tan (- \frac{π}{4}) |})}{8}$

Step 7.11.14

Multiply $2$ by $265.45698874$ .

$\frac{530.91397749 + \ln (\frac{| \sec (1.52734543) + \tan (1.52734543) |}{| \sec (- \frac{π}{4}) + \tan (- \frac{π}{4}) |})}{8}$

Step 7.11.15

$\sec (1.52734543) + \tan (1.52734543)$ is approximately $46.02172886$ which is positive so remove the absolute value

$\frac{530.91397749 + \ln (\frac{\sec (1.52734543) + \tan (1.52734543)}{| \sec (- \frac{π}{4}) + \tan (- \frac{π}{4}) |})}{8}$

Step 7.11.16

Simplify the denominator.

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Step 7.11.16.1

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{530.91397749 + \ln (\frac{\sec (1.52734543) + \tan (1.52734543)}{| \sec (\frac{7 π}{4}) + \tan (- \frac{π}{4}) |})}{8}$

Step 7.11.16.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\frac{530.91397749 + \ln (\frac{\sec (1.52734543) + \tan (1.52734543)}{| \sec (\frac{π}{4}) + \tan (- \frac{π}{4}) |})}{8}$

Step 7.11.16.3

The exact value of $\sec (\frac{π}{4})$ is $\frac{2}{\sqrt{2}}$ .

$\frac{530.91397749 + \ln (\frac{\sec (1.52734543) + \tan (1.52734543)}{| \frac{2}{\sqrt{2}} + \tan (- \frac{π}{4}) |})}{8}$

Step 7.11.16.4

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{530.91397749 + \ln (\frac{\sec (1.52734543) + \tan (1.52734543)}{| \frac{2}{\sqrt{2}} + \tan (\frac{7 π}{4}) |})}{8}$

Step 7.11.16.5

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because tangent is negative in the fourth quadrant.

$\frac{530.91397749 + \ln (\frac{\sec (1.52734543) + \tan (1.52734543)}{| \frac{2}{\sqrt{2}} - \tan (\frac{π}{4}) |})}{8}$

Step 7.11.16.6

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$\frac{530.91397749 + \ln (\frac{\sec (1.52734543) + \tan (1.52734543)}{| \frac{2}{\sqrt{2}} - 1 \cdot 1 |})}{8}$

Step 7.11.16.7

Multiply $- 1$ by $1$ .

$\frac{530.91397749 + \ln (\frac{\sec (1.52734543) + \tan (1.52734543)}{| \frac{2}{\sqrt{2}} - 1 |})}{8}$

Step 7.11.16.8

Simplify each term.

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Step 7.11.16.8.1

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

$\frac{530.91397749 + \ln (\frac{\sec (1.52734543) + \tan (1.52734543)}{| \frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} - 1 |})}{8}$

Step 7.11.16.8.2

Combine and simplify the denominator.

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Step 7.11.16.8.2.1

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

$\frac{530.91397749 + \ln (\frac{\sec (1.52734543) + \tan (1.52734543)}{| \frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}} - 1 |})}{8}$

Step 7.11.16.8.2.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{530.91397749 + \ln (\frac{\sec (1.52734543) + \tan (1.52734543)}{| \frac{2 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}} - 1 |})}{8}$

Step 7.11.16.8.2.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{530.91397749 + \ln (\frac{\sec (1.52734543) + \tan (1.52734543)}{| \frac{2 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}} - 1 |})}{8}$

Step 7.11.16.8.2.4

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

$\frac{530.91397749 + \ln (\frac{\sec (1.52734543) + \tan (1.52734543)}{| \frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}} - 1 |})}{8}$

Step 7.11.16.8.2.5

Add $1$ and $1$ .

$\frac{530.91397749 + \ln (\frac{\sec (1.52734543) + \tan (1.52734543)}{| \frac{2 \sqrt{2}}{{\sqrt{2}}^{2}} - 1 |})}{8}$

Step 7.11.16.8.2.6

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

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Step 7.11.16.8.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{530.91397749 + \ln (\frac{\sec (1.52734543) + \tan (1.52734543)}{| \frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}} - 1 |})}{8}$

Step 7.11.16.8.2.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{530.91397749 + \ln (\frac{\sec (1.52734543) + \tan (1.52734543)}{| \frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}} - 1 |})}{8}$

Step 7.11.16.8.2.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{530.91397749 + \ln (\frac{\sec (1.52734543) + \tan (1.52734543)}{| \frac{2 \sqrt{2}}{2^{\frac{2}{2}}} - 1 |})}{8}$

Step 7.11.16.8.2.6.4

Cancel the common factor of $2$ .

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Step 7.11.16.8.2.6.4.1

Cancel the common factor.

$\frac{530.91397749 + \ln (\frac{\sec (1.52734543) + \tan (1.52734543)}{| \frac{2 \sqrt{2}}{2^{\frac{2}{2}}} - 1 |})}{8}$

Step 7.11.16.8.2.6.4.2

Rewrite the expression.

$\frac{530.91397749 + \ln (\frac{\sec (1.52734543) + \tan (1.52734543)}{| \frac{2 \sqrt{2}}{2^{1}} - 1 |})}{8}$

Step 7.11.16.8.2.6.5

Evaluate the exponent.

$\frac{530.91397749 + \ln (\frac{\sec (1.52734543) + \tan (1.52734543)}{| \frac{2 \sqrt{2}}{2} - 1 |})}{8}$

Step 7.11.16.8.3

Cancel the common factor of $2$ .

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Step 7.11.16.8.3.1

Cancel the common factor.

$\frac{530.91397749 + \ln (\frac{\sec (1.52734543) + \tan (1.52734543)}{| \frac{2 \sqrt{2}}{2} - 1 |})}{8}$

Step 7.11.16.8.3.2

Divide $\sqrt{2}$ by $1$ .

$\frac{530.91397749 + \ln (\frac{\sec (1.52734543) + \tan (1.52734543)}{| \sqrt{2} - 1 |})}{8}$

Step 7.11.16.9

$\sqrt{2} - 1$ is approximately $0.41421356$ which is positive so remove the absolute value

$\frac{530.91397749 + \ln (\frac{\sec (1.52734543) + \tan (1.52734543)}{\sqrt{2} - 1})}{8}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{530.91397749 + \ln (\frac{\sec (1.52734543) + \tan (1.52734543)}{\sqrt{2} - 1})}{8}$

Decimal Form:

$66.95305809 \dots$

Step 9