Math basics › Calculus II
Improper integrals
What makes one improper
A limit of integration is ∞, or the function blows up somewhere in the interval. The second is easy to miss: 1/x² between −1 and 1 blows up at 0.
Replace it with a limit
∫ from 1 to ∞ means the limit, as b → ∞, of ∫ from 1 to b. Integrate normally, then take the limit.
Converge or diverge
If the limit is a finite number, the integral converges to it. If it is infinite or does not exist, it diverges.
The one to know
∫ from 1 to ∞ of 1/xᵖ dx converges when p > 1 and diverges when p ≤ 1. Most comparisons come back to it.