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Integrate by parts xInx​

User Cuong Tran
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1 Answer

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27 votes

Answer:


\displaystyle \int {x \ln x} \, dx = (x^2)/(2) \bigg( \ln(x) - (1)/(2) \bigg) + C

General Formulas and Concepts:

Calculus

Differentiation

  • Derivatives
  • Derivative Notation

Integration

  • Integrals
  • [Indefinite Integrals] Integration Constant C

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

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

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

Explanation:

Step 1: Define

Identify


\displaystyle \int {x \ln x} \, dx

Step 2: Integrate Pt. 1

Identify variables for integration by parts using LIPET.

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

Step 3: Integrate Pt. 2

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

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

Unit: Integration

User Sloane
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