CalcRef

Techniques of Integration · Integration by Parts

Integration by Parts

∫u dv=uv−∫v du\int u\, dv = uv - \int v\, du

The reverse of the product rule. Choose u and dv from the integrand using the LIATE mnemonic (Logarithmic, Inverse trig, Algebraic, Trig, Exponential).

Worked examples

Find ∫ x eˣ dx.
  1. u = x, dv = eˣ dx → du = dx, v = eˣ
  2. uv - ∫ v du = xeˣ - ∫ eˣ dx = xeˣ - eˣ + C

Answer: eˣ(x - 1) + C

Find ∫ x² sin x dx.
  1. u = x², dv = sin x dx → du = 2x dx, v = -cos x
  2. -x² cos x + ∫ 2x cos x dx
  3. Apply IBP again: u = 2x, dv = cos x dx → du = 2dx, v = sin x
  4. -x² cos x + 2x sin x - ∫ 2 sin x dx = -x² cos x + 2x sin x + 2 cos x + C

Answer: -x² cos x + 2x sin x + 2 cos x + C

Related formulas

Tabular Method (Repeated IBP)

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