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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 :

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Answer:

[tex]\displaystyle \int {x^3e^{x^4}} \, dx = \frac{e^{x^4}}{4} + C[/tex]

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

U-Substitution

Step-by-step explanation:

Step 1: Define

Identify

[tex]\displaystyle \int {x^3e^{x^4}} \, dx[/tex]

Step 2: Integrate Pt. 1

Identify variables for u-substitution.

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

Step 3: Integrate Pt. 2

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

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

Unit: Integration

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