Integration by Parts

Integration by Parts

Key Formula

• Integration by parts formula: ∫ u dv = uv - ∫ v du
• Purpose: Transforms product integral into simpler form by differentiating u and integrating dv
• Choice of u and dv: Usually choose u as polynomial or logarithmic, dv as trigonometric or exponential

Key Examples

• ∫ x sin(x) dx:
  • u = x, dv = sin(x) dx → du = dx, v = -cos(x)
  • Result: -x cos(x) + sin(x) + C
• ∫ x² e^x dx:
  • Apply twice: u = x², dv = e^x dx → first gives x²e^x - 2∫ x e^x dx
  • Second: u = x, dv = e^x dx → final: x²e^x - 2xe^x + 2e^x + C

Applications

• Fourier series: Compute coefficients by integrating products of functions (e.g., f(x) sin(nx) or f(x) cos(nx))
• Probability: Derive moments or expected values of continuous distributions (e.g., ∫ x f(x) dx for expectation)