Overview
This lecture introduces integration by parts, explains its connection to the product rule, derives the formula, and demonstrates its application through various examples, including special cases and "circular" integrals.
Derivation and Formula for Integration by Parts
- Integration by parts is based on the product rule for differentiation.
- The formula is: ∫u dv = uv − ∫v du.
- Choose u (to differentiate) and dv (to integrate); find du and v accordingly.
- Integration by parts gives you another integral; it is useful only if the resulting integral is simpler.
Choosing u and dv
- Pick u so that differentiating it simplifies the expression (polynomials are good choices).
- dv should be something you can integrate easily.
- Poor choices can lead to more complicated integrals.
Worked Examples
- ∫x sinx dx: Choose u = x (du = dx), dv = sinx dx (v = -cosx); result: -x cosx + sinx + c.
- ∫x eˣ dx: u = x, dv = eˣ dx; result: x eˣ − eˣ + c.
- ∫t² cos t dt: u = t², dv = cos t dt; apply integration by parts twice to reduce powers.
- ∫ln x dx: u = ln x, dv = dx; result: x ln x − x + c.
- ∫arctan x dx: u = arctan x, dv = dx; leads to a u-substitution for the resulting integral: x arctan x − (1/2) ln(1 + x²) + c.
- ∫ln x / x³ dx: u = ln x, dv = x⁻³ dx; leads to simple powers of x in the reduction.
Circular Integrals (Repeated Integrals)
- For ∫eˣ sin x dx: Applying integration by parts twice returns the original integral.
- Algebraically solve for the integral: I = ∫eˣ sin x dx leads to I = (eˣ(sin x − cos x))/2 + c.
Key Terms & Definitions
- Integration by Parts — a technique for integrating products of functions using ∫u dv = uv − ∫v du.
- u-substitution — a method for simplifying integrals by substituting variables.
- Product Rule — the derivative of uv is u dv + v du.
- Circular (or Recursive) Integral — an integral where repeated application of integration by parts cycles back to the original integral.
Action Items / Next Steps
- Practice integration by parts with polynomial, exponential, and trigonometric function combinations.
- Review and solve additional circular integral problems for mastery.
- Complete assigned homework problems on integration by parts.