Respuesta :

Space

Answer:

[tex]\displaystyle \int\limits^1_0 {\ln(3x)} \, dx = \ln(3) - 1[/tex]

General Formulas and Concepts:

Calculus

Differentiation

  • Derivatives
  • Derivative Notation

Derivative Property [Multiplied Constant]:                                                           [tex]\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)[/tex]

Basic Power Rule:

  1. f(x) = cxⁿ
  2. f’(x) = c·nxⁿ⁻¹

Derivative Rule [Chain Rule]:                                                                                 [tex]\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]

Integration

  • Integrals

Integration Rule [Reverse Power Rule]:                                                               [tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]

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

Integration Property [Multiplied Constant]:                                                         [tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]

Integration by Parts:                                                                                               [tex]\displaystyle \int {u} \, dv = uv - \int {v} \, du[/tex]

  • [IBP] LIPET: Logs, inverses, Polynomials, Exponentials, Trig

Step-by-step explanation:

Step 1: Define

Identify

[tex]\displaystyle \int\limits^1_0 {\ln(3x)} \, dx[/tex]

Step 2: Integrate Pt. 1

Identify variables for integration by parts using LIPET.

  1. Set u:                                                                                                             [tex]\displaystyle u = \ln(3x)[/tex]
  2. [u] Logarithmic Differentiation [Derivative Rule - Chain Rule]:                   [tex]\displaystyle du = \frac{(3x)'}{3x} \ dx[/tex]
  3. [du] Basic Power Rule [Derivative Property - Multiplied Constant]:           [tex]\displaystyle du = \frac{3}{3x} \ dx[/tex]
  4. [du] Simplify:                                                                                                   [tex]\displaystyle du = \frac{1}{x} \ dx[/tex]
  5. Set dv:                                                                                                           [tex]\displaystyle dv = 1 \ dx[/tex]
  6. [dv] Integration Rule [Reverse Power Rule]:                                               [tex]\displaystyle v = x[/tex]

Step 3: Integrate Pt. 2

  1. [Integral] Integration by Parts:                                                                         [tex]\displaystyle \int\limits^1_0 {\ln(3x)} \, dx = x \ln(3x) \bigg| \limits^1_0 - \int\limits^1_0 {1} \, dx[/tex]
  2. [Integral] Rewrite [Integration Property - Multiplied Constant]:                 [tex]\displaystyle \int\limits^1_0 {\ln(3x)} \, dx = x \ln(3x) \bigg| \limits^1_0 - \int\limits^1_0 {} \, dx[/tex]
  3. [Integral] Integration Rule [Reverse Power Rule]:                                       [tex]\displaystyle \int\limits^1_0 {\ln(3x)} \, dx = x \ln(3x) \bigg| \limits^1_0 - x \bigg| \limits^1_0[/tex]
  4. Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:           [tex]\displaystyle \int\limits^1_0 {\ln(3x)} \, dx = \ln(3) - 1[/tex]

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

Unit: Integration

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