Evaluating Improper Integrals BC only

An integral is improper when the interval is infinite or the integrand is unbounded on it, and both cases are handled by replacing the offending endpoint with a variable and taking a limit: $\int_{1}^{\infty} f = \lim_{b\to\infty}\int_{1}^{b} f$. If that limit is a finite number the integral converges to it, and otherwise it diverges. The limit notation is not a formality, since $F(\infty) - F(1)$ substitutes a symbol that is not a number, and it carries its own mark on a free-response question.

The integrand tending to zero is necessary and not sufficient: $\int_{1}^{\infty}\frac{dx}{x}$ diverges while $\int_{1}^{\infty}\frac{dx}{x^{2}} = 1$. The p-test says $\int_{1}^{\infty}x^{-p}dx$ converges exactly when $p > 1$ and $\int_{0}^{1}x^{-p}dx$ exactly when $p < 1$, the two halves pointing opposite ways because a slow tail and a steep spike are opposite dangers. A discontinuity strictly inside the interval announces itself nowhere in the notation: $\int_{-1}^{1}\frac{dx}{x^{2}}$ evaluates to $-2$ if handled carelessly, which is negative for a positive integrand. Split at the bad point and require both pieces to converge.