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Answer:

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

General Formulas and Concepts:

Calculus

Differentiation

  • Derivatives
  • Derivative Notation

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

Basic Power Rule:

  1. f(x) = cxⁿ
  2. f’(x) = c·nxⁿ⁻¹

Integration

  • Integrals
  • [Indefinite Integrals] Integration Constant C

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

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.

  1. Set u:                                                                                                             \(\displaystyle u = x\)
  2. [u] Differentiate [Derivative Rule - Basic Power Rule]:                               \(\displaystyle du = dx\)
  3. Set dv:                                                                                                           \(\displaystyle dv = e^{-5x} \ dx\)
  4. [dv] Integrate [Exponential Integration, U-Substitution]:                             \(\displaystyle v = \frac{-e^{-5x}}{5}\)

Step 3: Integrate Pt. 2

  1. [Integral] Integration by parts:                                                                       \(\displaystyle \int {xe^{-5x}} \, dx = \frac{-xe^{-5x}}{5} - \int {\frac{-e^{-5x}}{5}} \, dx\)
  2. [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.

  1. Set u:                                                                                                             \(\displaystyle u = -5x\)
  2. [u] Differentiate [Derivative Property, Basic Power Rule]:                        \(\displaystyle du = -5 \ dx\)

Step 5: Integrate Pt. 4

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