Answer:
[tex]\displaystyle \int {(x + 6)e^x} \, dx = (x + 5)e^x + C[/tex]
General Formulas and Concepts:
Calculus
Differentiation
- Derivatives
- Derivative Notation
Derivative Property [Addition/Subtraction]: [tex]\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)][/tex]
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Integration
- Integrals
- [Indefinite Integrals] Integration Constant C
Integration Property [Multiplied Constant]: [tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]
Integration by Parts: [tex]\displaystyle \int {u} \, dv = uv - \int {v} \, du[/tex]
- [IBP] LIPET: Logs, inverses, Polynomials, Exponentials, Trig
Step-by-step explanation:
Step 1: Define
Identify
[tex]\displaystyle \int {(x + 6)e^x} \, dx[/tex]
Step 2: Integrate Pt. 1
Identify variables for integration by parts using LIPET.
- Set u: [tex]\displaystyle u = x + 6[/tex]
- [u] Basic Power Rule [Derivative Property - Addition/Subtraction]: [tex]\displaystyle du = dx[/tex]
- Set dv: [tex]\displaystyle dv = e^x \ dx[/tex]
- [dv] Exponential Integration: [tex]\displaystyle v = e^x[/tex]
Step 3: Integrate Pt. 2
- [Integral] Integration by Parts: [tex]\displaystyle \int {(x + 6)e^x} \, dx = (x + 6)e^x - \int {e^x} \, dx[/tex]
- [Integral] Exponential Integration: [tex]\displaystyle \int {(x + 6)e^x} \, dx = (x + 6)e^x - e^x + C[/tex]
- Factor: [tex]\displaystyle \int {(x + 6)e^x} \, dx = e^x(x + 6 - 1) + C[/tex]
- Simplify: [tex]\displaystyle \int {(x + 6)e^x} \, dx = (x + 5)e^x + C[/tex]
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Integration
