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integrate by parts xInx​

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Answer:

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

Step 2: Integrate Pt. 1

Identify variables for integration by parts using LIPET.

  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 \ dx[/tex]
  4. [dv] Integration Rule [Reverse Power Rule]:                                               [tex]\displaystyle v = \frac{x^2}{2}[/tex]

Step 3: Integrate Pt. 2

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

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

Unit: Integration

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