Improper Integrals Lecture Notes

Introduction

• Topic: Improper Integrals
• Presented by: Jeetendra Kumar
• Platform: Maths by Jeetendra Kumar YouTube Channel

Key Definitions

Intervals

• Interval: A set defined as (x) such that (a \leq x \leq b).
  • Closed Interval: Denoted as ([a, b]), includes (a) and (b).
  • Open Interval: Denoted as ((a, b)), does not include (a) and (b).
  • Left Half Open Interval: ([a, b))
  • Right Half Open Interval: ((a, b])

Infinite Intervals

• Definition: An interval with infinite length (range) is an infinite interval.
• Examples: ((a, \infty)), ((-\infty, b)), ((-\infty, \infty))

Bounded and Unbounded Functions

• Bounded Function: (f(x)) is bounded over interval (I) if it fits within real numbers (A) and (B).
• Unbounded Function: Becomes infinite at some point within the interval.

Monotonic Functions

• Monotonically Increasing Function: For (x > y) implies (f(x) > f(y)).
• Monotonically Decreasing Function: For (x > y) implies (f(x) < f(y)).

Proper Integrals

• Definition: A definite integral (\int_a^b f(x) dx) is proper if:
  • Range of integration is finite.
  • Integrand (f(x)) is bounded.
• Examples:
  • (\int_0^5 \frac{1}{x+1} dx) : Finite range and bounded.
  • (\int_0^{\pi/2} \sin x dx) : Finite range and bounded.

Improper Integrals

• Definition: A definite integral is improper if:
  1. Interval is not finite but function is bounded.
  2. Interval is finite but function is unbounded.
  3. Neither interval is finite nor function is bounded.
• Examples:
  • (\int_0^\infty \frac{x}{x^2+1} dx) : Infinite range, bounded function.
  • (\int_0^5 \frac{1}{x-1} dx) : Finite range, unbounded function.
  • (\int_0^\infty \frac{1}{x(x-5)(x+2)} dx) : Infinite range, unbounded function.

Conclusion

• Wrap-up: Understanding of proper and improper integrals helps in evaluating integrals with infinite limits or unbounded functions.
• Further Study: For deeper understanding and live classes, refer to the PCM Guru application.