Proving Integrals Are Unbounded- Mathematical Approach

What "Unbounded" Actually Means

An integral is unbounded when its value grows without limit—or doesn't exist as a finite number. This isn't some abstract concept. It means the integral diverges, fails to converge, or approaches infinity.

Most students first encounter this with the classic example:

∫₁^∞ (1/x) dx = ∞

The area under y = 1/x from 1 to infinity is infinite. That's an unbounded integral. But the real skill is proving this for any integral—not just recognizing obvious cases.

The Core Criteria for Unboundedness

Before proving anything, you need to know what you're actually testing. An integral is unbounded when:

• The interval of integration is infinite (improper integral of Type 1)
• The integrand has a vertical asymptote within the interval (improper integral of Type 2)
• The limit defining the integral fails to exist as a finite number

Both conditions require the same proof strategy: show the limit doesn't converge to a finite value.

The Comparison Test: Your Main Weapon

This is the workhorse of unboundedness proofs. If you can compare your integral to a known divergent integral, you're done.

The Basic Logic

For improper integrals over [a, ∞):

• If f(x) ≥ g(x) ≥ 0 for all x ≥ a, and ∫g(x)dx diverges, then ∫f(x)dx also diverges
• You don't need to evaluate anything. You just need inequalities and a reference integral you already know diverges

The comparison test works because infinite + finite = infinite. If your function stays above a divergent baseline, your integral follows it into infinity.

Limit Comparison Test: The Refined Version

When direct comparison gets messy, use limits:

If lim(x→∞) f(x)/g(x) = c where c is finite and positive, then ∫f and ∫g either both converge or both diverge.

This is easier because you only need asymptotic behavior, not inequality throughout the interval.

The p-Test: Your Reference Standard

Every unboundedness proof eventually boils down to comparing against a p-integral. Know these cold:

• ∫₁^∞ (1/x^p) dx converges if p > 1, diverges if p ≤ 1
• ∫₀^1 (1/x^p) dx converges if p < 1, diverges if p ≥ 1

These two facts handle 90% of unboundedness proofs you'll encounter. The p-test IS the benchmark.

Direct Integration: When Comparison Isn't Enough

Sometimes you need to actually evaluate the limit. For functions with elementary antiderivatives, compute:

lim(b→∞) ∫ₐᵇ f(x) dx

If this limit is infinite or doesn't exist, the integral is unbounded. The classic example:

∫₁^b (1/x) dx = ln(b) - ln(1) = ln(b)

As b → ∞, ln(b) → ∞. Therefore, unbounded.

Testing for Unboundedness: A Practical Checklist

• Identify the type: infinite interval or infinite discontinuity?
• Set up the improper limit notation correctly
• Apply the comparison test or direct integration
• Compare to the p-test benchmark
• State your conclusion with justification

Comparison of Methods

Getting Started: A Worked Proof

Let's prove ∫₂^∞ (x² + 3x + 1)/(x³ + x) dx is unbounded.

Step 1: Simplify the asymptotic behavior

For large x, the dominant terms are x²/x³ = 1/x. So the integrand behaves like 1/x.

Step 2: Apply the Limit Comparison Test

Let f(x) = (x² + 3x + 1)/(x³ + x) and g(x) = 1/x.

lim(x→∞) f(x)/g(x) = lim(x→∞) [(x² + 3x + 1)/(x³ + x)] × [x/1]

= lim(x→∞) (x³ + 3x² + x)/(x³ + x) = 1

Step 3: Conclude

The limit is 1 (finite and positive). Since ∫₂^∞ 1/x dx diverges (p = 1 ≤ 1), our original integral also diverges. Unbounded.

Common Mistakes That Kill Proofs

• Wrong inequality direction: Comparing to a smaller divergent function doesn't work. Your function must be larger than or equal to the divergent benchmark
• Ignoring negative values: Comparison tests require non-negative functions. Handle absolute values or split intervals for signed functions
• Misidentifying the problem: Sometimes an integral diverges due to oscillation (like ∫sin(x)dx), not magnitude. Comparison tests won't help there
• Forgetting to check both directions: For finite intervals with asymptotes, check both one-sided limits

When You Should Actually Evaluate

Comparison tests are efficient, but sometimes direct computation is faster—especially for simple rational functions, exponentials, and basic logarithms. If the antiderivative is straightforward, just compute the limit. Don't force a comparison test when simple integration works.

The Bottom Line

Proving an integral is unbounded comes down to three moves: compare to a known divergent integral, compute the limit directly, or use the p-test as your benchmark. The comparison test is your default tool. Direct integration is for clean cases. The p-test is your reference point.

Master these three approaches and you can handle any unboundedness proof that comes your way.