Evaluating ∫₀³ dx/(x − 1)²

Calculus 2 · Axiom Academy

Finding the blow-up hidden inside the interval — and seeing what the Fundamental Theorem hands you if you miss it. Evaluate , or show that it diverges. Nicely done. The hard part of that problem was not the integration — it was noticing that there was something to notice. The integrand is positive everywhere it exists, and both halves of the interval enclose infinite area. Scan first, integrate second: an integral is improper if a limit of integration is infinite or the integrand blows up anywhere on the interval — including strictly inside it, where it is easiest to miss. Split, never straddle: a discontinuity at an interior point c forces , with the two limits taken independently. If either one fails, the whole integral diverges. An impossible answer is data: straight-through evaluation returned , a negative number for a strictly positive integrand. When a result contradicts something you know must be true, suspect a violated hypothesis rather than your arithmetic. Every convergence test in the series unit begins with this same reflex: find where the object misbehaves, isolate that point inside a limit, and only then compute.

This is the written version of the interactive lesson above. See the full Calculus 2 course.