Verified Answer :

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Answer:

\(\displaystyle 2 + \int\limits^6_2 {g(x)} \, dx = 18\)

General Formulas and Concepts:

Calculus

Integration

  • Integrals

Integration Property [Flipping Integral]:                                                               \(\displaystyle \int\limits^b_a {f(x)} \, dx = -\int\limits^a_b {f(x)} \, dx\)

Integration Property [Splitting Integral]:                                                               \(\displaystyle \int\limits^c_a {f(x)} \, dx = \int\limits^b_a {f(x)} \, dx + \int\limits^c_b {f(x)} \, dx\)

Step-by-step explanation:

Step 1: Define

Identify

\(\displaystyle \int\limits^8_2 {g(x)} \, dx = 13\)

\(\displaystyle \int\limits^8_6 {g(x)} \, dx = -3\)

\(\displaystyle 2 + \int\limits^6_2 {g(x)} \, dx\)

Step 2: Integrate

  1. [Integral] Rewrite [Integration Property - Flipping Integral]:                       \(\displaystyle \int\limits^8_6 {g(x)} \, dx = -3 \rightarrow \int\limits^6_8 {g(x)} \, dx = 3\)
  2. [Integral] Rewrite [Integration Property - Splitting Integral]:                       \(\displaystyle 2 + \int\limits^6_2 {g(x)} \, dx = 2 + \int\limits^8_2 {g(x)} \, dx + \int\limits^6_8 {g(x)} \, dx\)
  3. [Integrals] Substitute:                                                                                    \(\displaystyle 2 + \int\limits^6_2 {g(x)} \, dx = 2 + 13 + 3\)
  4. Simplify:                                                                                                         \(\displaystyle 2 + \int\limits^6_2 {g(x)} \, dx = 18\)

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

Unit: Integration