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Evaluate the integral using integration by parts with the indicated choices of u and dv. (Use C for the constant of integration.) xe7x dx; u

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Answer:

[tex]\displaystyle \int {xe^{7x}} \, dx = \frac{e^{7x}}{7} \bigg( x - \frac{1}{7} \bigg) + C[/tex]

General Formulas and Concepts:

Calculus

Differentiation

  • Derivatives
  • Derivative Notation

Derivative Property [Multiplied Constant]:                                                           [tex]\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)[/tex]

Basic Power Rule:

  1. f(x) = cxⁿ
  2. f’(x) = c·nxⁿ⁻¹

Integration

  • Integrals

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

U-Substitution

Integration by Parts:                                                                                               [tex]\displaystyle \int {u} \, dv = uv - \int {v} \, du[/tex]

  • [IBP] LIPET: Logs, inverses, Polynomials, Exponentials, Trig

Step-by-step explanation:

Step 1: Define

Identify

[tex]\displaystyle \int {xe^{7x}} \, dx[/tex]

Step 2: Integrate Pt. 1

Identify variables for integration by parts using LIPET.

  1. Set u:                                                                                                             [tex]\displaystyle u = x[/tex]
  2. [u] Basic Power Rule:                                                                                     [tex]\displaystyle du = dx[/tex]
  3. Set dv:                                                                                                           [tex]\displaystyle dv = e^{7x} \ dx[/tex]
  4. [dv] Exponential Integration [U-Substitution]:                                             [tex]\displaystyle v = \frac{e^{7x}}{7}[/tex]

Step 3: integrate Pt. 2

  1. [Integral] Integration by Parts:                                                                      [tex]\displaystyle \int {xe^{7x}} \, dx = \frac{xe^{7x}}{7} - \int {\frac{e^{7x}}{7}} \, dx[/tex]
  2. [Integral] Rewrite [Integration Property - Multiplied Constant]:                 [tex]\displaystyle \int {xe^{7x}} \, dx = \frac{xe^{7x}}{7} - \frac{1}{7} \int {e^{7x}} \, dx[/tex]

Step 4: Integrate Pt. 3

Identify variables for u-substitution.

  1. Set u:                                                                                                             [tex]\displaystyle u = 7x[/tex]
  2. [u] Basic Power Rule [Derivative Property - Multiplied Constant]:             [tex]\displaystyle du = 7 \ dx[/tex]

Step 5: Integrate Pt. 4

  1. [Integral] Rewrite [Integration Property - Multiplied Constant]:                 [tex]\displaystyle \int {xe^{7x}} \, dx = \frac{xe^{7x}}{7} - \frac{1}{49} \int {7e^{7x}} \, dx[/tex]
  2. [Integral] U-Substitution:                                                                               [tex]\displaystyle \int {xe^{7x}} \, dx = \frac{xe^{7x}}{7} - \frac{1}{49} \int {e^u} \, dx[/tex]
  3. [Integral] Exponential Integration:                                                               [tex]\displaystyle \int {xe^{7x}} \, dx = \frac{xe^{7x}}{7} - \frac{e^u}{49} + C[/tex]
  4. [u] Back-Substitute:                                                                                       [tex]\displaystyle \int {xe^{7x}} \, dx = \frac{xe^{7x}}{7} - \frac{e^{7x}}{49} + C[/tex]
  5. Factor:                                                                                                           [tex]\displaystyle \int {xe^{7x}} \, dx = \frac{e^{7x}}{7} \bigg( x - \frac{1}{7} \bigg) + C[/tex]

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

Unit: Integration

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