Respuesta :

Space

Answer:

g(x) = 4x² - x

General Formulas and Concepts:

Algebra II

Functions

  • Function Notation
  • Piecewise Functions

Calculus

Integration

  • Integrals
  • Integral Notation
  • Integration Constant C

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

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

Integration Property [Addition/Subtraction]:                                                       [tex]\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx[/tex]

Step-by-step explanation:

*Note:

Integrating a piecewise function requires you to integrate both parts.

Step 1: Define

Identify.

[tex]\displaystyle f(x) = \left \{ {{8x - 1 ,\ x \leq 4} \atop {31 ,\ x \geq 4}} \right.[/tex]

[tex]\displaystyle \int {f(x)} \, dx = \left \{ {{g(x) + C ,\ x \leq 4} \atop {31x + C ,\ x \geq 4}} \right.[/tex]

Step 2: Find function g(x)

We can see that the 2nd part of the piecewise function already has been integrated:

  1. [Integral] Set up:                                                                                             [tex]\displaystyle \int {f(x)} \, dx ,\ x \geq 4 = \int {31} \, dx ,\ x \geq 4[/tex]
  2. [Integral] Rewrite [Integration Property - Multiplied Constant]:                 [tex]\displaystyle \int {f(x)} \, dx ,\ x \geq 4 = 31 \int {} \, dx ,\ x \geq 4[/tex]
  3. [Integral] Integrate [Integration Rule - Reverse Power Rule]:                     [tex]\displaystyle \int {f(x)} \, dx ,\ x \geq 4 = 31x + C ,\ x \geq 4[/tex]

To find function g(x), we simply have the same setup:

  1. [Integral] Set up:                                                                                             [tex]\displaystyle \int {f(x)} \, dx ,\ x \leq 4 = \int {8x - 1} \, dx ,\ x \leq 4[/tex]
  2. [Integral] Rewrite [Integration Rule - Addition/Subtraction]:                       [tex]\displaystyle \int {f(x)} \, dx ,\ x \leq 4 = \int {8x} \, dx - \int {1} \, dx ,\ x \leq 4[/tex]
  3. [Integrals] Rewrite [Integration Property - Multiplied Constant]:               [tex]\displaystyle \int {f(x)} \, dx ,\ x \leq 4 = 8 \int {x} \, dx - \int {} \, dx ,\ x \leq 4[/tex]
  4. [Integrals] Integrate [Integration Rule - Reverse Power Rule]:                   [tex]\displaystyle \int {f(x)} \, dx ,\ x \leq 4 = 8 \bigg( \frac{x^2}{2} \bigg) - x + C ,\ x \leq 4[/tex]
  5. Simplify:                                                                                                         [tex]\displaystyle \int {f(x)} \, dx ,\ x \leq 4 = 4x^2 - x + C ,\ x \leq 4[/tex]
  6. Redefine:                                                                                                       [tex]\displaystyle g(x) = 4x^2 - x + C ,\ x \leq 4[/tex]

The integration constant C is already included in the answer, so our answer is g(x) = 4x² - x.

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

Unit: Integration (Applications)