7.8, #4. ∫_{b}^{0} \frac{e^{x}}{1 + e^{x}} dx = ln(1 + e^{x})|_{b}^{0} = ln 2 − ln(1 + e^{b}). This has limit ln 2 as b → −∞.

7.8, #17. ∫_{2}^{b} \frac{1}{x ln x} dx = ln(ln x)|_{2}^{b} = ln(ln b) − ln(ln 2) (use the substitution w = ln x.) This diverges as b → ∞.

7.8, #20. ∫_{2}^{b} \frac{1}{x(ln x)^{2}} dx = −\frac{1}{ln x}|_{3}^{b} = \frac{1}{ln 3} − \frac{1}{ln b} (use the substitution w = ln x.) This has limit (1/ ln 3) as b → ∞.

7.8, #27. ∫ \frac{dz}{z^{2} − z} = ln |z − 1| − ln |z| + C = ln(|z − 1|/|z|) + C using partial fractions or formula 26 on page 367 with x = z, a = 1, b = 0. f(z) = 1/(z^{2} − z) has graph

0  1^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}

and is undefined at z = 0 and z = 1. So ∫_{a}^{b} f(z) dz is improper if 0 or 1 is in [a, b]. ln |z − 1| − ln |z| + C has no limit as z → 0 or as z → 1 from either the right or the left. Thus ∫_{a}^{x} f(z) dz diverges if 0 or 1 is in [a, x]. In particular, ∫_{1}^{3} f(z) dz diverges. If 0 and 1 are not in [a, x], then ∫_{a}^{x} f(z) dz = ln |x−1|−ln |x|−ln |a−1|+ln |a| = ln(|a||x−1|/|a−1||x|). Lastly,

∫_{3}^{∞} \frac{dz}{z^{2} − z} dz = lim_{b→∞} ln |1 − (1/b)| − ln(2/3) = ln(3/2).

7.8, #30. People get sick fastest when the rate r = 1000te^{−0.5t} is maximal. Since r′ = 1000e^{−0.5t}(1 − .5t) is zero when t = 2, this happens 2 days after the start of the epidemic. The total number of people getting sick is 4000. Indeed,

∫_{0}^{∞} 1000te^{−0.5t} dt = lim_{b→∞} ∫_{0}^{b} 1000te^{−0.5t} dt

= lim_{b→∞} 1000(−2t − 4)e^{−0.5t}|_{0}^{b}

= lim_{b→∞} 1000((−2b − 4)e^{−.5b} + 4)

= 4000

7.9, #3–4. The integrals converge. Indeed,

\begin{aligned}∫_{1}^{∞} \frac{dx}{x^{3} + 1} & < ∫_{1}^{∞} \frac{dx}{x^{3}} \\ ∫_{1}^{∞} \frac{dθ}{\sqrt{θ^{3} + 1}} & < ∫_{1}^{∞} \frac{dθ}{θ^{3/2}} \end{aligned}

and the integrals on the right converge since the integrands are of the form 1/x^{p} with p > 1.

7.9, #5. The integral diverges since

∫_{1}^{b} \frac{dx}{1 + x} = ln |1 + x||_{1}^{b} = ln |1 + b| − ln 2

diverges when b → ∞.

7.9, #15. θ^{3} + θ behaves like θ as θ → 0 since θ^{3} ≤ θ for 0 ≤ θ ≤ 1. So we would guess the integral converges. Indeed, θ^{3} + θ > θ, so

∫_{0}^{1} \frac{dθ}{\sqrt{θ^{3} + θ}} < ∫_{0}^{1} \frac{dθ}{\sqrt{θ}} .

The last integral converges since the integrand is of the form 1/θ^{p} with p < 1.