A student evaluates ∫₋∞^∞ x dx by noting that ∫₋ᵀ^ᵀ x dx = 0 for any T, and concludes the answer is 0. What is wrong with this reasoning?
ANothing — the integral equals 0 by the symmetry of an odd function
BThe definition requires lim_{s→−∞} and lim_{t→+∞} of ∫ₛᵗ x dx taken independently; since neither limit exists finitely, the integral diverges
CThe student should verify using the substitution u = −x before concluding
DThe integral converges, but the value is indeterminate rather than 0
Taking the symmetric limit lim_{T→∞} ∫₋ᵀ^ᵀ x dx = 0 computes the Cauchy principal value, which is a weaker notion than convergence. The proper definition requires two independent limits: lim_{s→−∞} and lim_{t→+∞} of ∫ₛᵗ x dx. Since ∫₀ᵗ x dx = t²/2 → ∞, the integral toward +∞ alone diverges. The integral ∫₋∞^∞ x dx is divergent; the symmetric computation masks this by exploiting cancellation that the definition does not permit.
Question 2 Multiple Choice
∫₁^∞ 1/xᵖ dx converges for p > 1, and ∫₀¹ 1/xᵖ dx converges for p < 1. A student claims these are contradictory — the same function 1/xᵖ cannot converge in opposite conditions. What is the correct response?
AThe student is right — both integrals actually share the same convergence condition
BThe conditions are complementary: the first probes decay at infinity, the second probes blow-up near 0; each has its own convergence requirement
CThe student is correct that only p between 0 and 1 makes both integrals converge simultaneously
DThe formulas apply only to integer values of p, so the comparison is invalid
The two integrals ask different questions. ∫₁^∞ 1/xᵖ dx asks whether 1/xᵖ decays fast enough at infinity — it does when p > 1. ∫₀¹ 1/xᵖ dx asks whether the blow-up near x = 0 is integrable — it is when p < 1 (so the singularity is mild enough). These are independent conditions about different behaviors of the same function. Working through both cases is the standard way to build intuition for improper integral convergence.
Question 3 True / False
The improper integral ∫₁^∞ sin(x)/x dx diverges because the integrand does not approach 0 fast enough.
TTrue
FFalse
Answer: False
This integral converges — it is the classic example of conditional convergence. sin(x)/x oscillates with decreasing amplitude and does approach 0, and the alternating cancellation makes the integral converge despite the fact that ∫₁^∞ |sin(x)/x| dx diverges. Conditional convergence is real: an improper integral can converge without converging absolutely, just as an alternating series can converge while the series of absolute values diverges.
Question 4 True / False
The two limits in ∫₋∞^∞ f(x) dx must be taken independently; combining them into the symmetric limit lim_{T→∞} ∫₋ᵀ^ᵀ f(x) dx gives the Cauchy principal value, which may differ from the true integral value.
TTrue
FFalse
Answer: True
This is precisely the subtlety the definition is designed to capture. The Cauchy principal value exploits symmetric cancellation that may not persist when the limits are taken independently. For odd functions like f(x) = x, the principal value is 0 but the integral diverges. For integrals that converge by the proper definition, the principal value agrees — but the definition must come first.
Question 5 Short Answer
Explain why defining ∫ₐ^∞ f(x) dx as lim_{t→∞} ∫ₐᵗ f(x) dx, rather than 'plugging in ∞,' is necessary rather than a mere formality.
Think about your answer, then reveal below.
Model answer: Because ∞ is not a number and cannot be substituted. The limit formulation is essential for detecting when the integral diverges, for identifying conditional convergence, and for correctly handling doubly improper integrals by keeping the two independent limits separate. Without the limit framework, symmetric cancellation can make a divergent integral appear to have a finite value — the Cauchy principal value — which is a different and weaker notion than genuine convergence.
The limit definition is not bureaucratic precision for its own sake. It provides the only rigorous way to determine whether a definite value exists, and it exposes the difference between absolute and conditional convergence. Skipping the limit process and 'plugging in ∞' works for simple cases but breaks down on examples like ∫₋∞^∞ x dx, where the naive answer of 0 conceals a divergent integral.