Improper Integrals with an Infinite Discontinuity

The Fundamental Theorem of Calculus lets you evaluate an integral by plugging endpoints into an antiderivative, but that shortcut assumes the function is continuous everywhere on the interval. If the curve shoots off to infinity at some point c in [a, b], the region under it near that point might still have finite area, or it might not. You cannot tell just by plugging in, because the antiderivative itself is often undefined or discontinuous right at that point. You have to check carefully, using a limit.

The fix is to rewrite the integral as a limit. If the discontinuity sits at the upper limit b, replace b with a variable t, integrate from a to t, and take the limit as t approaches b from the left. If the discontinuity is at the lower limit, do the same from the right. If it sits inside the interval, split the integral into two pieces at that point and take two separate limits, one for each side; the whole thing only converges if both limits exist and are finite. Then evaluate using the antiderivative on the safe, finite side, where it stays continuous, and let the limit variable creep toward the asymptote.

This topic is where integrals meet the limit ideas from earlier units, and it sets up the convergence tests you use on improper integrals with infinite bounds. The same instinct, replace the dangerous value with a letter and take a limit, carries forward into comparing series and integrals for convergence.