Question

Exercise is mixed —some require integration by parts, while others can be integrated by using techniques discussed in the chapter on Integration.

∫x^2 e^2x dx.

Answer 1

`\displaystyle \int {x^2e^(2x)} \, dx = (e^(2x))/(2) \bigg( x^2 - x + (1)/(2) \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:

*Note:

You can use tabular integration instead of integrating by parts twice.

Step 1: Define

Identify

`\displaystyle \int {x^2e^(2x)} \, dx`

Step 2: Integrate Pt. 1

Identify variables for integration by parts using LIPET.

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

Step 3: Integrate Pt. 2

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

Step 4: Integrate Pt. 3

Identify variables for integration by parts using LIPET.

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

Step 5: Integrate Pt. 4

1. [Integral] Integration by Parts:
   `\displaystyle \int {x^2e^(2x)} \, dx = (x^2e^(2x))/(2) - \bigg( (xe^(2x))/(2) - \int {(e^(2x))/(2)} \, dx \bigg)`
2. [Integral] Rewrite [Integration Property - Multiplied Constant]:
   `\displaystyle \int {x^2e^(2x)} \, dx = (x^2e^(2x))/(2) - \bigg( (xe^(2x))/(2) - (1)/(2) \int {e^(2x)} \, dx \bigg)`

Step 6: Integrate Pt. 5

Identify variables for u-substitution.

1. Set u:
   `\displaystyle u = 2x`
2. [u] Basic Power Rule [Derivative Property - Multiplied Constant]:
   `\displaystyle du = 2 \ dx`

Step 7: Integrate Pt. 6

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

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

Unit: Integration