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Techniques of Integration · Integration by Parts

Tabular Method (Repeated IBP)

Alternating signs: (+)u⋅v1−u′⋅v2+u′′⋅v3−⋯\text{Alternating signs: } (+)u \cdot v_1 - u' \cdot v_2 + u'' \cdot v_3 - \cdots

A shortcut for repeated integration by parts when one factor is a polynomial. Create a table of derivatives and antiderivatives with alternating signs.

Worked examples

Find ∫ x³ eˣ dx using the tabular method.
  1. Derivatives of x³: x³, 3x², 6x, 6, 0
  2. Integrals of eˣ: eˣ, eˣ, eˣ, eˣ
  3. Signs: +, -, +, -
  4. Result: x³eˣ - 3x²eˣ + 6xeˣ - 6eˣ + C

Answer: eˣ(x³ - 3x² + 6x - 6) + C

Related formulas

Integration by Parts

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