Question

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

Answer 1

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

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Multiplied Constant]:
`\displaystyle (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

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 = (-e^(-5x))/(5)`

Step 3: Integrate Pt. 2

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

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

Unit: Integration