Respuesta :

Answer:

-93

Step-by-step explanation:

This is just.a matter of using a couple of integration rules and plugging in then using order of operations.

Difference rule and constant multiple rule will be used here.

3(-17)-7(6)

-51-42

-93

Space

Answer:

[tex]\displaystyle \int\limits^6_4 {[3f(x) - 7g(x)]} \, dx = -93[/tex]

General Formulas and Concepts:

Calculus

Integration

  • Integrals

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:

Step 1: Define

Identify

[tex]\displaystyle \int\limits^6_4 {f(x)} \, dx = -17[/tex]

[tex]\displaystyle \int\limits^6_4 {g(x)} \, dx = 6[/tex]

[tex]\displaystyle \int\limits^6_4 {[3f(x) - 7g(x)]} \, dx[/tex]

Step 2: Integrate

  1. [Integral] Rewrite [Integration Property - Addition/Subtraction]:               [tex]\displaystyle \int\limits^6_4 {[3f(x) - 7g(x)]} \, dx = \int\limits^6_4 {3f(x)} \, dx - \int\limits^6_4 {7g(x)} \, dx[/tex]
  2. [Integrals] Rewrite [Integration Property - Multiplied Constant]:              [tex]\displaystyle \int\limits^6_4 {[3f(x) - 7g(x)]} \, dx = 3 \int\limits^6_4 {f(x)} \, dx - 7 \int\limits^6_4 {g(x)} \, dx[/tex]
  3. [Integrals] Substitute:                                                                                    [tex]\displaystyle \int\limits^6_4 {[3f(x) - 7g(x)]} \, dx = 3(-17) - 7(6)[/tex]
  4. Simplify:                                                                                                         [tex]\displaystyle \int\limits^6_4 {[3f(x) - 7g(x)]} \, dx = -93[/tex]

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

Unit: Integration

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