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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.) 2x2 ln(x) dx; u = ln(x), dv = 2x2 dx

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Answer:

[tex]\displaystyle \int {2x^2 \ln x} \, dx = \frac{2x^3 \big( 3 \ln(x) - 1 \big)}{9} + C[/tex]

General Formulas and Concepts:

Calculus

Differentiation

  • Derivatives
  • Derivative Notation

Integration

  • Integrals
  • [Indefinite Integrals] Integration Constant C

Integration Rule [Reverse Power Rule]:                                                               [tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]

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

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 {2x^2 \ln x} \, dx[/tex]

Step 2: Integrate Pt. 1

  1. [Integral] Rewrite [Integration Property - Multiplied Constant]:                 [tex]\displaystyle \int {2x^2 \ln x} \, dx = 2 \int {x^2 \ln x} \, dx[/tex]

Step 3: integrate Pt. 2

Identify variables for integration by parts using u-substitution.

  1. Set u:                                                                                                             [tex]\displaystyle u = \ln(x)[/tex]
  2. [u] Logarithmic Differentiation:                                                                    [tex]\displaystyle du = \frac{1}{x} \ dx[/tex]
  3. Set dv:                                                                                                           [tex]\displaystyle dv = x^2 \ dx[/tex]
  4. [dv] Integration Rule [Reverse Power Rule]:                                               [tex]\displaystyle v = \frac{x^3}{3}[/tex]

Step 4: Integrate Pt. 3

  1. [Integral] Integration by Parts:                                                                       [tex]\displaystyle \int {2x^2 \ln x} \, dx = 2 \bigg( \frac{x^3 \ln(x)}{3} - \int {\frac{x^2}{3}} \, dx \bigg)[/tex]
  2. [Integral] Rewrite [Integration Property - Multiplied Constant]:                 [tex]\displaystyle \int {2x^2 \ln x} \, dx = 2 \bigg( \frac{x^3 \ln(x)}{3} - \frac{1}{3} \int {x^2} \, dx \bigg)[/tex]
  3. [Integral] Integration Rule [Reverse Power Rule]:                                       [tex]\displaystyle \int {2x^2 \ln x} \, dx = 2 \bigg( \frac{x^3 \ln(x)}{3} - \frac{x^3}{9} \bigg) + C[/tex]
  4. Simplify:                                                                                                         [tex]\displaystyle \int {2x^2 \ln x} \, dx = \frac{2x^3 \big( 3 \ln(x) - 1 \big)}{9} + C[/tex]

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

Unit: Integration

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