Question

Use integration by parts to find the integrals in Exercise.
∫(x+6)ex dx.

Answer 1

`\displaystyle \int {(x + 6)e^x} \, dx = (x + 5)e^x + C`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Addition/Subtraction]:
`\displaystyle (d)/(dx)[f(x) + g(x)] = (d)/(dx)[f(x)] + (d)/(dx)[g(x)]`

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Integration

• Integrals
• [Indefinite Integrals] Integration Constant C

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

Integration by Parts:
`\displaystyle \int {u} \, dv = uv - \int {v} \, du`

• [IBP] LIPET: Logs, inverses, Polynomials, Exponentials, Trig

Explanation:

Step 1: Define

Identify

`\displaystyle \int {(x + 6)e^x} \, dx`

Step 2: Integrate Pt. 1

Identify variables for integration by parts using LIPET.

1. Set u:
   `\displaystyle u = x + 6`
2. [u] Basic Power Rule [Derivative Property - Addition/Subtraction]:
   `\displaystyle du = dx`
3. Set dv:
   `\displaystyle dv = e^x \ dx`
4. [dv] Exponential Integration:
   `\displaystyle v = e^x`

Step 3: Integrate Pt. 2

1. [Integral] Integration by Parts:
   `\displaystyle \int {(x + 6)e^x} \, dx = (x + 6)e^x - \int {e^x} \, dx`
2. [Integral] Exponential Integration:
   `\displaystyle \int {(x + 6)e^x} \, dx = (x + 6)e^x - e^x + C`
3. Factor:
   `\displaystyle \int {(x + 6)e^x} \, dx = e^x(x + 6 - 1) + C`
4. Simplify:
   `\displaystyle \int {(x + 6)e^x} \, dx = (x + 5)e^x + C`

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

Unit: Integration