Indefinite Integral Worksheet- Comparison Test Explained

What the Comparison Test Actually Is

The Comparison Test is a tool for determining whether an improper integral converges or diverges without calculating its exact value. You compare your target integral to one you already know something about.

That's it. That's the whole idea.

Most improper integrals don't have nice antiderivatives you can write down. The Comparison Test lets you sidestep that problem entirely by checking your integral against something simpler.

When This Test Actually Works

You use the Comparison Test when:

• Your integrand looks complicated and doesn't integrate nicely
• You can find a function that's either always larger or always smaller than your integrand
• You know whether the simpler function converges or diverges

If you can't find a comparable function, this test won't help you. Pick a different approach.

The Two Versions You Need to Know

Direct Comparison Test

If 0 ≤ f(x) ≤ g(x) on [a, ∞), then:

• If ∫g converges, then ∫f also converges
• If ∫f diverges, then ∫g also diverges

The key insight: a smaller function that's bounded is more likely to converge. A larger function that diverges forces the smaller one to diverge too.

Limit Comparison Test

When direct comparison gets messy, take a limit:

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

This version is easier to apply because you only need the ratio to approach a non-zero constant. The exact value doesn't matter.

How to Actually Apply This

Step 1: Identify Your Integrand

Write down f(x) clearly. Know whether you're dealing with an integral from a to ∞, or a bounded interval with an infinite discontinuity.

Step 2: Find a Comparison Function

Look at the dominant behavior of f(x) as x approaches infinity. Drop lower-order terms.

Examples:

• f(x) = (x² + 3x)/(x³ + 1) → compare to 1/x
• f(x) = e^(-x²) → compare to e^(-x)
• f(x) = sin(x)/x² → compare to 1/x²

Step 3: Check the Inequality or Limit

For direct comparison: verify 0 ≤ f ≤ g or 0 ≤ g ≤ f across your interval.

For limit comparison: compute lim f(x)/g(x) and confirm it's a positive finite number.

Step 4: Apply the Test and Conclude

Match your integrand's behavior to the known behavior of your comparison function.

Worked Examples

Example 1: Direct Comparison

Determine if ∫₁^∞ (1)/(x³ + 1) dx converges.

For x ≥ 1, we have x³ + 1 > x³, so 1/(x³ + 1) < 1/x³.

We know ∫₁^∞ 1/x³ dx converges (p-integral with p = 3 > 1).

Since our integrand is smaller than a convergent integral, the original integral converges.

Example 2: Limit Comparison

Determine if ∫₂^∞ (x²)/(x³ + 5x) dx converges.

Dominant terms: x²/x³ = 1/x. Let g(x) = 1/x.

Compute the limit: lim (x→∞) [x²/(x³ + 5x)] / [1/x] = lim (x→∞) x³/(x³ + 5x) = 1.

Since L = 1 is finite and positive, and ∫₂^∞ 1/x dx diverges, the original integral diverges.

Quick Reference: Comparison Functions

These are the standard functions you should compare against:

Common Mistakes That Will Cost You Points

• Forgetting to check positivity: Both functions must be non-negative on the interval. The test fails otherwise.
• Choosing a comparison that's too close: If g(x) - f(x) is too small, you might miss divergence.
• Mismatching p-values: Make sure you're comparing to the right power of x.
• Wrong direction on the inequality: Double-check: is f ≤ g or g ≤ f? The conclusion changes depending on which way you go.
• Ignoring the limit test's conditions: The limit must be finite and positive, not zero or infinity.

Practice: Getting Started

Test each integral using either direct or limit comparison:

1. ∫₁^∞ (x)/(x³ + 2x + 1) dx
2. ∫₃^∞ (1)/(√x + 5) dx
3. ∫₁^∞ e^(-x²) dx
4. ∫₂^∞ (sin²x)/(x²) dx

For #1: Compare to 1/x². For #2: Compare to 1/√x. For #3: Compare to e^(-x). For #4: Compare to 1/x² and use the fact that sin²x ≤ 1.

When to Pick a Different Test

The Comparison Test isn't always your best option. Consider alternatives:

• Absolute convergence test: For integrals with oscillating functions like sin(x) or cos(x)
• Integration by parts: When you can actually solve the integral but need to handle limits carefully
• Ratio or root tests: For integrals that look like infinite series disguised as integrals

Comparison works best when your integrand has a clear dominant term you can isolate. If the function bounces around wildly or has no obvious simpler comparison, try something else.