Question

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.

Answer 1

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]:
`\displaystyle \int {x^n} \, dx = (x^(n + 1))/(n + 1) + 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

Explanation:

*Note:

Treat a and n as arbitrary constants.

Step 1: Define

Identify

`\displaystyle \int {x^ne^(ax)} \, dx`

Step 2: Integrate Pt. 1

Identify variables for integration by parts using LIPET.

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

Step 3: Integrate Pt. 2

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

∴ We have verified/derived the formula.

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

Unit: Integration