Improper Integral

An improper integral has at least one of:

1. Infinite limit: $\int_a^\infty f(x) \, dx$ or $\int_{-\infty}^\infty f(x) \, dx$.
2. Unbounded integrand somewhere in $[a, b]$ (vertical asymptote).

Both are evaluated as limits of proper integrals:

$\int_a^\infty f(x) \, dx = \lim_{b \to \infty} \int_a^b f(x) \, dx$

If finite, converges; otherwise diverges.

Famous examples:

• $\int_1^\infty \frac{1}{x^2} dx = 1$ ✓
• $\int_1^\infty \frac{1}{x} dx = \infty$ ✗ (slower decay diverges)
• $\int_{-\infty}^\infty e^{-x^2} dx = \sqrt{\pi}$ — Gaussian integral.

Convergence tests (comparison, p-test) decide whether to bother integrating. Improper integrals appear in probability (PDF normalisation), Fourier transforms, and physics.