- Topics At a Glance
- Indefinite Integrals Introduction
- Integration by Substitution: Indefinite Integrals
- Legrange (Prime) Notation
- Leibniz (Fraction) Notation
- Integration by Substitution: Definite Integrals
- Integration by Parts: Indefinite Integrals
- Some Tricks
- Integration by Parts: Definite Integrals
- Integration by Partial Fractions
- Integrating Definite Integrals
- Choosing an Integration Method
- Integration by Substitution
- Integration by Parts
- Integration by Partial Fractions
- Thinking Backwards
- Improper Integrals
- Badly Behaved Limits
- Badly Behaved Functions
- Badly Behaved Everything
- Comparing Improper Integrals
- The p-Test
- Finite and Infinite Areas
- Comparison with Formulas
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem
Introduction to :
We often use integrals of the functions 
, for various values of p, to help determine whether other integrals converge or diverge.
You already did the work to show this, so we'll just summarize the results. Assuming p is greater than 0 (because otherwise the exponents do weird things),
converges if p < 1 and diverges otherwise.
converges if p > 1 and diverges otherwise.
This is often called the p-test for improper integrals.
Practice:
Example 1
Use the <em>p</em>-test to determine if
converges or diverges. |
where p > 1 and the interval of integration is [0,1], this integral diverges.
gets close to the x-axis quickly as x approaches ∞. If you can remember this, then you can remember that
Exercise 1
Use the p-test to determine if the integral converges or diverges.
with p > 1 and we're integrating from 0 to 1.Exercise 2
Use the p-test to determine if the integral converges or diverges.
with p < 1, so this integral converges.Exercise 3
Use the p-test to determine if the integral converges or diverges.
with p > 1 and we're integrating from 1 to ∞.Exercise 4
Use the p-test to determine if the integral converges or diverges.
is of the form
