replace it with a variable bound
∞CALCULUS II STUDY GUIDE
Improper Integrals
Evaluate infinite intervals and unbounded integrands with limits, comparison, and p-integral tests.
8 key ideas8 flashcards5 practice questions
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Every troublesome bound gets its own limit.
split into one-sided pieces
all required limits must exist
Key ideas to know
Start with the relationships between ideas. Then close the notes and explain each one from memory.
- 01
An infinite bound is replaced by a finite bound and a limit.
- 02
An unbounded integrand requires a one-sided limit at the discontinuity.
- 03
An interior discontinuity splits an integral into two separate improper integrals.
- 04
Both pieces must converge for the full integral to converge.
- 05
The integral of 1/xᵖ from 1 to infinity converges exactly when p is greater than one.
- 06
The integral of 1/xᵖ from 0 to 1 converges exactly when p is less than one.
- 07
Direct comparison uses inequalities between nonnegative functions.
- 08
Limit comparison uses a finite positive ratio to transfer convergence behavior.
PART OF THE CALCULUS II GUIDE
See every set in this course and follow a focused review order.
Open the full Calculus II guide →Two ideas worth correcting now
NOT QUITE
→A bounded area picture proves convergence
USE THIS INSTEAD
Only the defining limit decides convergence.
NOT QUITE
→Positive and negative divergent pieces may be canceled
USE THIS INSTEAD
The standard improper integral requires each piece to converge separately.
Flashcards
Answer before opening each card. The effort to retrieve is part of the learning.
1What replaces an upper bound of infinity?Show answer ＋
A variable bound b followed by the limit as b tends to infinity.
2What must happen at an interior discontinuity?Show answer ＋
Split the integral and test both one-sided pieces.
3When does ∫₁∞1/xᵖ dx converge?Show answer ＋
When p > 1.
4When does ∫₀¹1/xᵖ dx converge?Show answer ＋
When p < 1.
5What does divergence mean here?Show answer ＋
The defining limit is not a finite real number.
6What conditions fit direct comparison?Show answer ＋
Both functions are nonnegative on the tested interval.
7What ratio fits limit comparison?Show answer ＋
A finite positive limit of f(x)/g(x).
8Can cancellation rescue two divergent pieces?Show answer ＋
No. Each improper piece must converge on its own.
Explain it in your own words
Use the answer as a check after you have written or spoken your response.
01Does ∫₁∞1/x² dx converge?
Yes. It is a p-integral with p = 2, and its result is 1.
02Why does ∫₁∞1/x dx diverge?
Its antiderivative is ln x, which grows without bound.
03How is ∫₋₁¹1/x² dx treated?
Split at zero; both one-sided integrals diverge, so the full integral diverges.
04How can 1/(x²+1) be tested on an infinite interval?
Compare it with 1/x² for large x or evaluate with arctangent.
05Why is writing infinity directly into an antiderivative invalid?
Infinity is not a number; the result is defined through a limit.
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