Question

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

∫x^3ex^4 dx.

Answer 1

`\displaystyle \int {x^3e^(x^4)} \, dx = (e^(x^4))/(4) + C`

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

U-Substitution

Explanation:

Step 1: Define

Identify

`\displaystyle \int {x^3e^(x^4)} \, dx`

Step 2: Integrate Pt. 1

Identify variables for u-substitution.

1. Set u:
   `\displaystyle u = x^4`
2. [u] Basic Power Rule:
   `\displaystyle du = 4x^3 \ dx`

Step 3: Integrate Pt. 2

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

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

Unit: Integration