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Use integration by parts to derive the following formula from the table of integrals.
∫x^n . e^ax dx=x^ne^ax/a-n/a∫x^n-1e^ax dx +C, a≠-0.

Respuesta :

Space

Answer:

See explanation.

General Formulas and Concepts:

Calculus

Differentiation

  • Derivatives
  • Derivative Notation

Basic Power Rule:

  1. f(x) = cxⁿ
  2. f’(x) = c·nxⁿ⁻¹

Integration

  • Integrals
  • [Indefinite Integrals] Integration Constant C

Integration Rule [Reverse Power Rule]:                                                               [tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]

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

U-Substitution

Integration by Parts:                                                                                               [tex]\displaystyle \int {u} \, dv = uv - \int {v} \, du[/tex]

  • [IBP] LIPET: Logs, inverses, Polynomials, Exponentials, Trig

Step-by-step explanation:

*Note:

Treat a and n as arbitrary constants.

Step 1: Define

Identify

[tex]\displaystyle \int {x^ne^{ax}} \, dx[/tex]

Step 2: Integrate Pt. 1

Identify variables for integration by parts using LIPET.

  1. Set u:                                                                                                             [tex]\displaystyle u = x^n[/tex]
  2. [u] Basic Power Rule:                                                                                     [tex]\displaystyle du = nx^{n - 1} \ dx[/tex]
  3. Set dv:                                                                                                           [tex]\displaystyle dv = e^{ax} \ dx[/tex]
  4. [dv] Exponential Integration [U-Substitution]:                                             [tex]\displaystyle v = \frac{e^{ax}}{a}[/tex]

Step 3: Integrate Pt. 2

  1. [Integral] Integration by Parts:                                                                       [tex]\displaystyle \int {x^ne^{ax}} \, dx = \frac{x^na^{ax}}{a} - \int {\frac{nx^{n - 1}e^{ax}}{a}} \, dx[/tex]
  2. [Integral] Rewrite [Integration Property - Multiplied Constant]:                 [tex]\displaystyle \int {x^ne^{ax}} \, dx = \frac{x^na^{ax}}{a} - \frac{n}{a} \int {x^{n - 1}e^{ax}} \, dx ,\ a \neq 0[/tex]

∴ We have verified/derived the formula.

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

Unit: Integration

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