Respuesta :

Space

Answer:

[tex]\displaystyle \int\limits^7_1 {\frac{1}{x}} \, dx = \ln 7[/tex]

General Formulas and Concepts:

Calculus

Integration

  • Integrals

Integration Rule [Fundamental Theorem of Calculus 1]:                                     [tex]\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)[/tex]

Area of a Region Formula:                                                                                     [tex]\displaystyle A = \int\limits^b_a {[f(x) - g(x)]} \, dx[/tex]

Step-by-step explanation:

Step 1: Define

Identify

[tex]\displaystyle f(x) = \frac{1}{x} \\\left[ 1 ,\ 7 \right][/tex]

Step 2: Integrate

  1. Substitute in variables [Area of a Region Formula]:                                   [tex]\displaystyle \int\limits^7_1 {\frac{1}{x}} \, dx[/tex]
  2. [Integral] Logarithmic Integration:                                                               [tex]\displaystyle \int\limits^7_1 {\frac{1}{x}} \, dx = \ln \big| x \big| \bigg| \limits^7_1[/tex]
  3. Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:           [tex]\displaystyle \int\limits^7_1 {\frac{1}{x}} \, dx = \ln 7 - \ln 1[/tex]
  4. Simplify:                                                                                                         [tex]\displaystyle \int\limits^7_1 {\frac{1}{x}} \, dx = \ln 7[/tex]

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

Unit: Integration

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