Question

Please show work it’s for calc

Answer 1

-93

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

Answer 2

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

General Formulas and Concepts:

Calculus

Integration

• Integrals

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

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

Explanation:

Step 1: Define

Identify

`\displaystyle \int\limits^6_4 {f(x)} \, dx = -17`

`\displaystyle \int\limits^6_4 {g(x)} \, dx = 6`

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

Step 2: Integrate

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

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

Unit: Integration