The Improper Integral Calculator evaluates integrals whose interval is infinite or whose integrand becomes unbounded at a point. These integrals must be interpreted as limits rather than ordinary finite-interval integrals.
Types of improper integrals
An integral such as β«ββ 1/xΒ² dx has an infinite upper limit. An integral such as β«βΒΉ 1/βx dx has an unbounded integrand at x = 0. In both cases, a limit determines whether the integral converges.
Worked examples
For β«ββ 1/xΒ² dx, evaluate the limit lim(bββ) β«βα΅ xβ»Β² dx = 1, so the integral converges. By contrast, β«ββ 1/x dx diverges because the corresponding limit grows without bound.
Checking convergence
Identify every infinite endpoint or singularity, split the integral when necessary, and evaluate the relevant limits separately. A finite limit means convergence; failure to approach a finite value means divergence.