Respuesta :

Space

Answer:

(a)  [tex]\displaystyle \int {\frac{\sec x \tan x}{1 + \sec x}} \, dx = \boxed{ \ln | 1 + \sec x | + C }[/tex]

General Formulas and Concepts:
Calculus

Differentiation

  • Derivatives
  • Derivative Notation

Derivative Property [Addition/Subtraction]:
[tex]\displaystyle (u + v)' = u' + v'[/tex]
Derivative Rule [Basic Power Rule]:

  1. f(x) = cxⁿ
  2. f’(x) = c·nxⁿ⁻¹

Integration

  • Integrals

Integration Methods: U-Substitution and U-Solve

Step-by-step explanation:

Step 1: Define

Identify given.

[tex]\displaystyle \int {\frac{\sec x \tan x}{1 + \sec x}} \, dx[/tex]

Step 2: Integrate Pt. 1

Identify variables for u-substitution/u-solve.

  1. Set u:
    [tex]\displaystyle u = 1 + \sec x[/tex]
  2. [u] Differentiate [Derivative Rules, Properties, and Trigonometric Differentiation]:
    [tex]\displaystyle du = \sec x \tan x \ dx[/tex]
  3. [du] Rewrite [U-Solve]:
    [tex]\displaystyle dx = \cos x \cot x \ du[/tex]

Step 3: Integrate Pt. 2

  1. [Integral] Apply Integration Method [U-Solve]:
    [tex]\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}[/tex]
  2. [Integrand] Simplify:
    [tex]\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}[/tex]
  3. [Integral] Apply Logarithmic Integration:
    [tex]\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}[/tex]
  4. [u] Back-substitute:
    [tex]\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}[/tex]

∴ we have used substitution to find the indefinite integral.

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Learn more about Calculus: https://brainly.com/question/27746481

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Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Integration

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