Expii

# 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.