What Is an Improper Integral? - Expii

Earlier, we used Riemann sums to define definite integrals of nice (usually continuous or piecewise-continuous), bounded functions \(f(x)\) on bounded intervals \([a,b]\). However, sometimes we have an integral where the interval is unbounded (in other words, one of the endpoints \(a,b\) is infinite), or the integrand is unbounded (in other words, \(f(x)\) has an infinite singularity on the interval \([a,b]\)), or possibly both. In these situations, the integral \(\int_a^b f(x)\,dx\) is called an improper integral, which can still be properly interpreted and computed as a sum of limits of definite integrals.