Comparison test (for improper integrals)

Bound an unfamiliar integrand above or below by a known one to determine convergence without evaluating.

The comparison test determines an improper integral's convergence WITHOUT evaluating it directly: if $0\le f(x)\le g(x)$ and $\int g$ converges, then $\int f$ converges too (squeezed from above); if $0\le g(x)\le f(x)$ and $\int g$ diverges, then $\int f$ diverges too (dragged up from below). It's useful exactly when $f$ itself has no elementary antiderivative but a simpler bounding function does.

Examples

$\int_1^{\infty}\frac{1}{x^2+1}\,dx$ converges by comparison, since $\frac{1}{x^2+1}\le\frac{1}{x^2}$ and $\int_1^{\infty}\frac{1}{x^2}\,dx$ converges.