Improper Integrals: Introduction with Solved Examples

Ordinary definite integrals need a finite interval and a well-behaved function. But science constantly asks about infinite horizons — the total energy radiated forever, the probability over all time. An improper integral extends integration to these cases: infinite limits, or integrands that blow up at a point, handled by taking a limit of ordinary integrals. The integral converges if that limit is finite, and diverges otherwise.

This matters because the most important integrals in probability and physics are improper: the normal distribution integrates from −∞ to ∞, and decay processes integrate to infinity. Knowing when such integrals converge — and computing them — is essential beyond school calculus.

Key Concepts

• Type 1 — infinite limits: ∫ₐ^∞ f(x) dx = lim(b→∞) ∫ₐᵇ f(x) dx.
• Type 2 — infinite integrand: if f blows up at c in [a, b], split at c and take limits from each side.
• Convergence vs divergence: the limit must exist and be finite; otherwise the integral diverges (has no value).
• p-test: ∫₁^∞ dx/xᵖ converges if p > 1 and diverges if p ≤ 1 — the boundary case ∫₁^∞ dx/x diverges.
• Split at every bad point: an integral with two trouble spots must be split into pieces that are each handled separately.

Formulas

• ∫ₐ^∞ f(x) dx = lim(b→∞) ∫ₐᵇ f(x) dx
• ∫₁^∞ dx/x² = 1 (converges); ∫₁^∞ dx/x diverges
• ∫₀^∞ e^(−x) dx = 1
• ∫₀¹ dx/√x = 2 (integrand blows up at 0, but the integral converges)

Worked Example

Evaluate ∫₁^∞ dx/x². Step 1: replace ∞ with a variable limit b: ∫₁ᵇ x⁻² dx. Step 2: integrate normally: [−1/x]₁ᵇ = −1/b + 1 = 1 − 1/b. Step 3: let b → ∞: 1/b → 0, so the limit is 1. The integral converges to 1. Contrast with ∫₁^∞ dx/x = lim(b→∞) ln b = ∞, which diverges — the power 2 versus 1 makes all the difference, exactly as the p-test predicts.

Common Misconceptions

• Plugging in ∞ like a number: ∞ is not a number; you must write the limit explicitly and evaluate it.
• Assuming blow-ups always diverge: ∫₀¹ dx/√x converges to 2 even though 1/√x → ∞ at 0. The spike is narrow enough that the area stays finite.
• Ignoring interior bad points: ∫₋₁¹ dx/x² diverges because of x = 0 inside the interval — you cannot just apply the Fundamental Theorem across it.

Real-World Applications

Every continuous probability distribution (normal, exponential) is normalised by an improper integral over an infinite range. Physicists compute total radiated energy, escape velocities and field energies with integrals to infinity. In finance, the present value of a perpetual income stream is an improper integral.

Improper integrals are ordinary integrals plus a limit. Write the limit, evaluate, and check convergence — and remember the p-test: for 1/xᵖ at infinity, only p > 1 converges.