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julie426

21.01.2022

Mathematics

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

General Formulas and Concepts:

Calculus

Differentiation

DerivativesDerivative Notation

Derivative Property [Addition/Subtraction]: $\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]$

Basic Power Rule:

f(x) = cxⁿf’(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

Step-by-step 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.

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

Step 3: Integrate Pt. 2

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

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

Unit: Integration

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