Respuesta :

Space

Answer:

[tex]\displaystyle \int\limits^5_1 {x^4(1 + 2x^5)} \, dx = \frac{9768748}{5}[/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]

Derivative Property [Addition/Subtraction]:                                                         [tex]\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)][/tex]  

Basic Power Rule:

  1. f(x) = cxⁿ
  2. f’(x) = c·nxⁿ⁻¹

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]

U-Substitution

Step-by-step explanation:

Step 1: Define

Identify

[tex]\displaystyle \int\limits^5_1 {x^4(1 + 2x^5)} \, dx[/tex]

Step 2: Integrate Pt. 1

Identify variables for u-substitution.

  1. Set u:                                                                                                             [tex]\displaystyle u = 1 + 2x^5[/tex]
  2. [u] Differentiate [Basic Power Rule, Derivative Properties]:                       [tex]\displaystyle du = 10x^4 \ dx[/tex]
  3. [Bounds] Switch:                                                                                           [tex]\displaystyle \left \{ {{x = 5,\ u = 1 + 2(5)^5 = 6251} \atop {x=1,\ u = 1 + 2(1)^5 = 3}} \right.[/tex]

Step 3: Integrate Pt. 2

  1. [Integral] Rewrite [Integration Property - Multiplied Constant]:                 [tex]\displaystyle \int\limits^5_1 {x^4(1 + 2x^5)} \, dx = \frac{1}{10}\int\limits^5_1 {10x^4(1 + 2x^5)} \, dx[/tex]
  2. [Integral] U-Substitution:                                                                               [tex]\displaystyle \int\limits^5_1 {x^4(1 + 2x^5)} \, dx = \frac{1}{10}\int\limits^{6251}_3 {u} \, du[/tex]
  3. [Integral] Reverse Power Rule:                                                                     [tex]\displaystyle \int\limits^5_1 {x^4(1 + 2x^5)} \, dx = \frac{1}{10} \bigg( \frac{u^2}{2} \bigg) \bigg| \limits^{6251}_3[/tex]
  4. Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:           [tex]\displaystyle \int\limits^5_1 {x^4(1 + 2x^5)} \, dx = \frac{1}{10}(19537496)[/tex]
  5. Simplify:                                                                                                         [tex]\displaystyle \int\limits^5_1 {x^4(1 + 2x^5)} \, dx = \frac{9768748}{5}[/tex]

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

Unit: Integration

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