Mathematics, 14.01.2020 20:31, 2020seogang

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

Answers: 3

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answered: Kookiexperfect9420

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

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

answered: Guest

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This graph shows how much anthony earns babysitting, compared with the number of hours he works. sel

This graph shows how much anthony earns babysitting, compared with the number of hours he works. sel

answered: Guest

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