Improper integrals

The integral also extends to unbounded intervals, defined as the limit of an ordinary integral: $\int_a^{+\infty}f\,dx = \lim_{t\to+\infty}\int_a^t f\,dx.$ The improper integral converges if the limit is finite, otherwise it diverges.

A quick criterion much used concerns powers: $\int_1^\infty x^{-p}\,dx \quad\text{converge per } p>1,\quad\text{diverge per } p\le 1.$

Example — Two cases compared

$\int_1^\infty\frac{dx}{x^2} = \left[-\frac{1}{x}\right]_1^\infty = 0-(-1) = 1 \quad\text{(converge)}.$ $\int_1^\infty\frac{dx}{x} = [\ln x]_1^\infty = +\infty \quad\text{(diverge)}.$

The same scheme (integral as a limit) applies also when the function has a vertical asymptote at an endpoint of the domain.

Links

Topics: Integrale
Concepts: Integrale definito · Integrale improprio · Limite
Skills: Analisi casi limite · Calcolare limiti · Integrare