Tutorial 2: Integration by Parts

Tutorial problems focused on integration by parts technique.

Sections

Basic Integration by Parts (Problems 1-4)

  • Single application of parts formula
  • Choosing $u$ and $dv$ using LIATE rule
  • Integrals of the form $\int x^n e^x dx$, $\int x \ln x dx$

Repeated Integration by Parts (Problems 5-8)

  • Multiple applications required
  • Tabular method for efficiency
  • Cyclic integrals (e.g., $\int e^x \sin x dx$)

Special Cases (Problems 9-12)

  • Integrating $\ln x$ and inverse trig functions
  • Reduction formulas
  • Definite integrals using parts

LIATE Rule Reminder

Choose $u$ in order of preference:

  1. Logarithmic ($\ln x$)
  2. Inverse trig ($\arcsin x$, $\arctan x$)
  3. Algebraic ($x^n$)
  4. Trigonometric ($\sin x$, $\cos x$)
  5. Exponential ($e^x$)

Common Patterns

  • $\int x^n e^x dx$: repeated parts
  • $\int x^n \ln x dx$: $u = \ln x$
  • $\int e^{ax} \sin(bx) dx$: two applications, solve for integral

Links