Answer:
(a) ![\displaystyle \int {\frac{\sec x \tan x}{1 + \sec x}} \, dx = \boxed{ \ln | 1 + \sec x | + C }](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%7B%5Cfrac%7B%5Csec%20x%20%5Ctan%20x%7D%7B1%20%2B%20%5Csec%20x%7D%7D%20%5C%2C%20dx%20%3D%20%5Cboxed%7B%20%5Cln%20%7C%201%20%2B%20%5Csec%20x%20%7C%20%2B%20C%20%7D)
General Formulas and Concepts:
<u>Calculus</u>
Differentiation
- Derivatives
- Derivative Notation
Derivative Property [Addition/Subtraction]:
![\displaystyle (u + v)' = u' + v'](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%28u%20%2B%20v%29%27%20%3D%20u%27%20%2B%20v%27)
Derivative Rule [Basic Power Rule]:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Integration
Integration Methods: U-Substitution and U-Solve
Step-by-step explanation:
<u>Step 1: Define</u>
<em>Identify given.</em>
<em />![\displaystyle \int {\frac{\sec x \tan x}{1 + \sec x}} \, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint%20%7B%5Cfrac%7B%5Csec%20x%20%5Ctan%20x%7D%7B1%20%2B%20%5Csec%20x%7D%7D%20%5C%2C%20dx)
<u>Step 2: Integrate Pt. 1</u>
<em>Identify variables for u-substitution/u-solve</em>.
- Set <em>u</em>:
![\displaystyle u = 1 + \sec x](https://tex.z-dn.net/?f=%5Cdisplaystyle%20u%20%3D%201%20%2B%20%5Csec%20x)
- [<em>u</em>] Differentiate [Derivative Rules, Properties, and Trigonometric Differentiation]:
![\displaystyle du = \sec x \tan x \ dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20du%20%3D%20%5Csec%20x%20%5Ctan%20x%20%5C%20dx)
- [<em>du</em>] Rewrite [U-Solve]:
![\displaystyle dx = \cos x \cot x \ du](https://tex.z-dn.net/?f=%5Cdisplaystyle%20dx%20%3D%20%5Ccos%20x%20%5Ccot%20x%20%5C%20du)
<u>Step 3: Integrate Pt. 2</u>
- [Integral] Apply Integration Method [U-Solve]:
![\displaystyle \begin{aligned}\int {\frac{\sec x \tan x}{1 + \sec x}} \, dx & = \int {\frac{\sec x \tan x \cos x \cot x}{u}} \, du\end{aligned}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cbegin%7Baligned%7D%5Cint%20%7B%5Cfrac%7B%5Csec%20x%20%5Ctan%20x%7D%7B1%20%2B%20%5Csec%20x%7D%7D%20%5C%2C%20dx%20%26%20%3D%20%5Cint%20%7B%5Cfrac%7B%5Csec%20x%20%5Ctan%20x%20%5Ccos%20x%20%5Ccot%20x%7D%7Bu%7D%7D%20%5C%2C%20du%5Cend%7Baligned%7D)
- [Integrand] Simplify:
![\displaystyle \begin{aligned}\int {\frac{\sec x \tan x}{1 + \sec x}} \, dx & = \int {\frac{\sec x \tan x \cos x \cot x}{u}} \, du \\& = \int {\frac{1}{u}} \, du \\\end{aligned}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cbegin%7Baligned%7D%5Cint%20%7B%5Cfrac%7B%5Csec%20x%20%5Ctan%20x%7D%7B1%20%2B%20%5Csec%20x%7D%7D%20%5C%2C%20dx%20%26%20%3D%20%5Cint%20%7B%5Cfrac%7B%5Csec%20x%20%5Ctan%20x%20%5Ccos%20x%20%5Ccot%20x%7D%7Bu%7D%7D%20%5C%2C%20du%20%5C%5C%26%20%3D%20%5Cint%20%7B%5Cfrac%7B1%7D%7Bu%7D%7D%20%5C%2C%20du%20%5C%5C%5Cend%7Baligned%7D)
- [Integral] Apply Logarithmic Integration:
![\displaystyle \begin{aligned}\int {\frac{\sec x \tan x}{1 + \sec x}} \, dx & = \int {\frac{\sec x \tan x \cos x \cot x}{u}} \, du \\& = \int {\frac{1}{u}} \, du \\& = \ln | u | + C \\\end{aligned}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cbegin%7Baligned%7D%5Cint%20%7B%5Cfrac%7B%5Csec%20x%20%5Ctan%20x%7D%7B1%20%2B%20%5Csec%20x%7D%7D%20%5C%2C%20dx%20%26%20%3D%20%5Cint%20%7B%5Cfrac%7B%5Csec%20x%20%5Ctan%20x%20%5Ccos%20x%20%5Ccot%20x%7D%7Bu%7D%7D%20%5C%2C%20du%20%5C%5C%26%20%3D%20%5Cint%20%7B%5Cfrac%7B1%7D%7Bu%7D%7D%20%5C%2C%20du%20%5C%5C%26%20%3D%20%5Cln%20%7C%20u%20%7C%20%2B%20C%20%5C%5C%5Cend%7Baligned%7D)
- [<em>u</em>] Back-substitute:
![\displaystyle \begin{aligned}\int {\frac{\sec x \tan x}{1 + \sec x}} \, dx & = \int {\frac{\sec x \tan x \cos x \cot x}{u}} \, du \\& = \int {\frac{1}{u}} \, du \\& = \ln | u | + C \\& = \boxed{ \ln | 1 + \sec x | + C } \\\end{aligned}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cbegin%7Baligned%7D%5Cint%20%7B%5Cfrac%7B%5Csec%20x%20%5Ctan%20x%7D%7B1%20%2B%20%5Csec%20x%7D%7D%20%5C%2C%20dx%20%26%20%3D%20%5Cint%20%7B%5Cfrac%7B%5Csec%20x%20%5Ctan%20x%20%5Ccos%20x%20%5Ccot%20x%7D%7Bu%7D%7D%20%5C%2C%20du%20%5C%5C%26%20%3D%20%5Cint%20%7B%5Cfrac%7B1%7D%7Bu%7D%7D%20%5C%2C%20du%20%5C%5C%26%20%3D%20%5Cln%20%7C%20u%20%7C%20%2B%20C%20%5C%5C%26%20%3D%20%5Cboxed%7B%20%5Cln%20%7C%201%20%2B%20%5Csec%20x%20%7C%20%2B%20C%20%7D%20%5C%5C%5Cend%7Baligned%7D)
∴ we have used substitution to <em>find</em> the indefinite integral.
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Learn more about integration: brainly.com/question/27746485
Learn more about Calculus: brainly.com/question/27746481
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Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Integration
Answer:
Min: 38
Q1: 47
Q2: 56.5
Q3: 64
Max: 71
Step-by-step explanation:
64 38 50 71 60 58 47 55 70 45
Sort the data:
38 45 47 50 55 58 60 64 70 71
Min: 38
Q1: 47
Q2: (55+58)/2 = 56.5
Q3: 64
Max: 71