Mathematics: Evaluate the integral of xe^(-5x) dx using integrate by parts.

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Evaluate the integral of xe^(-5x) dx using integrate

Ответ:

rainbowcakes914

02.01.2022

Mathematics

$\displaystyle \int {xe^{-5x}} \, dx = -e^{-5x} \bigg( \frac{x}{5} + \frac{1}{25} \bigg) + C$

General Formulas and Concepts:

Calculus

Differentiation

DerivativesDerivative Notation

Derivative Property [Multiplied Constant]: $\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)$

Basic Power Rule:

f(x) = cxⁿf’(x) = c·nxⁿ⁻¹

Integration

Integrals[Indefinite Integrals] Integration Constant C

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

U-Substitution

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

$\displaystyle \int {xe^{-5x}} \, dx$

Step 2: Integrate Pt. 1

Identify variables for integration by parts using LIPET.

Set u: $\displaystyle u = x$ [u] Differentiate [Derivative Rule - Basic Power Rule]: $\displaystyle du = dx$ Set dv: $\displaystyle dv = e^{-5x} \ dx$ [dv] Integrate [Exponential Integration, U-Substitution]: $\displaystyle v = \frac{-e^{-5x}}{5}$

Step 3: Integrate Pt. 2

[Integral] Integration by parts: $\displaystyle \int {xe^{-5x}} \, dx = \frac{-xe^{-5x}}{5} - \int {\frac{-e^{-5x}}{5}} \, dx$ [Integral] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle \int {xe^{-5x}} \, dx = \frac{-xe^{-5x}}{5} + \frac{1}{5} \int {e^{-5x}} \, dx$

Step 4: Integrate Pt. 3

Identify variables for u-substitution.

Set u: $\displaystyle u = -5x$ [u] Differentiate [Derivative Property, Basic Power Rule]: $\displaystyle du = -5 \ dx$

Step 5: Integrate Pt. 4

[Integral] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle \int {xe^{-5x}} \, dx = \frac{-xe^{-5x}}{5} - \frac{1}{25} \int {-5e^{-5x}} \, dx$ [Integral] U-Substitution: $\displaystyle \int {xe^{-5x}} \, dx = \frac{-xe^{-5x}}{5} - \frac{1}{25} \int {e^u} \, du$ [Integral] Exponential Integration: $\displaystyle \int {xe^{-5x}} \, dx = \frac{-xe^{-5x}}{5} - \frac{e^u}{25} + C$ [u] Back-Substitute: $\displaystyle \int {xe^{-5x}} \, dx = \frac{-xe^{-5x}}{5} - \frac{e^{-5x}}{25} + C$ Factor: $\displaystyle \int {xe^{-5x}} \, dx = -e^{-5x} \bigg( \frac{x}{5} + \frac{1}{25} \bigg) + C$

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

Unit: Integration

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Evaluate the integral of xe^(-5x) dx using integrate by parts.

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