Revision sheet

Infinite intervals

1. Replace $\infty$ by $b$.
2. Integrate.
3. Take $\lim_{b \to \infty}$: a number means convergence.

Watch out: $f \to 0$ does not guarantee convergence.

Infinite discontinuities

• Look for asymptotes inside $[a, b]$ before integrating.
• Replace the bad end by $t$; take the one-sided limit.

Watch out: never apply the FTC across an asymptote.

The p-test

• $\int_1^{\infty} x^{-p}\,dx$: converges for $p > 1$.
• $\int_0^1 x^{-p}\,dx$: converges for $p < 1$.
• $p = 1$: both diverge.

Watch out: the condition flips between $\infty$ and $0$.