Chapter 17 — Exercises

38 problems on improper integrals: infinite limits, infinite integrands, convergence and divergence, the p-integral tests, comparison and limit comparison, the Gamma and Gaussian integrals, and applications across four fields. Work them with paper and pen first; reach for scipy only to check, never to replace, your reasoning.

Tier key: ⭐ routine · ⭐⭐ standard · ⭐⭐⭐ challenging · ⭐⭐⭐⭐ advanced.

Part A — Type 1: Infinite Limits of Integration (Section 17.1)

17.1 ⭐ Evaluate $\displaystyle\int_1^\infty \frac{1}{x^3}\,dx$.

17.2 ⭐ Evaluate $\displaystyle\int_0^\infty e^{-2x}\,dx$.

17.3 ⭐ Show that $\displaystyle\int_1^\infty \frac{1}{\sqrt{x}}\,dx$ diverges, and explain in one sentence why the decay of $1/\sqrt{x}$ is "too slow."

17.4 ⭐ Evaluate $\displaystyle\int_{-\infty}^0 e^{x}\,dx$.

17.5 ⭐⭐ Evaluate $\displaystyle\int_0^\infty x\,e^{-x^2}\,dx$ using the substitution $u = x^2$.

17.6 ⭐⭐ Evaluate $\displaystyle\int_{-\infty}^\infty \frac{1}{1+x^2}\,dx$ and confirm it equals $\pi$ (compare Worked Example 17.1.4 in the chapter).

17.7 ⭐⭐ Evaluate $\displaystyle\int_2^\infty \frac{1}{x\,(\ln x)^2}\,dx$ using $u = \ln x$. Does it converge? To what?

17.8 ⭐⭐ Show that $\displaystyle\int_2^\infty \frac{1}{x\,\ln x}\,dx$ diverges, again via $u = \ln x$. Contrast this verdict with Exercise 17.7 and explain the role of the extra $\ln x$.

17.9 ⭐⭐⭐ Evaluate $\displaystyle\int_0^\infty x\,e^{-x}\,dx$ by integration by parts, then confirm your answer against the Gamma-function value $\Gamma(2)$ (Section 17.4).

17.10 ⭐⭐⭐ The integral $\displaystyle\int_{-\infty}^\infty \frac{x}{1+x^2}\,dx$ diverges, yet its Cauchy principal value is $0$. Show both facts: that each half $\int_0^\infty$ and $\int_{-\infty}^0$ diverges, and that $\lim_{T\to\infty}\int_{-T}^T \frac{x}{1+x^2}\,dx = 0$ (Section 17.1, Section 17.7).

Part B — Type 2: Infinite Integrands (Section 17.2)

17.11 ⭐ Evaluate $\displaystyle\int_0^1 \frac{1}{\sqrt[3]{x}}\,dx$ (singularity at $x=0$).

17.12 ⭐ Evaluate $\displaystyle\int_0^1 \frac{1}{\sqrt{1-x}}\,dx$ (singularity at $x=1$).

17.13 ⭐⭐ Show that $\displaystyle\int_0^4 \frac{1}{x-2}\,dx$ diverges. Identify where the singularity sits and why blind FTC ("$\ln|x-2|$ evaluated at the endpoints") is illegitimate here (Section 17.2).

17.14 ⭐⭐ Evaluate $\displaystyle\int_0^1 \ln x\,dx$ using integration by parts. Confirm it converges, and to $-1$. (The integrand is unbounded as $x\to 0^+$.)

17.15 ⭐⭐ Evaluate $\displaystyle\int_{-1}^1 \frac{1}{x^{2/3}}\,dx$. The integrand blows up at the interior point $0$; split there and add the two pieces.

17.16 ⭐⭐⭐ Evaluate $\displaystyle\int_0^2 \frac{1}{(x-1)^{2/3}}\,dx$ (interior singularity at $x=1$). Then change one exponent so the integral diverges, and justify your choice using the p-integral-at-zero rule.

Part C — Convergence and Divergence by the p-Integral Tests (Sections 17.1, 17.2)

17.17 ⭐ Without computing any antiderivative, classify each as convergent or divergent and state the exponent $p$: (a) $\displaystyle\int_1^\infty x^{-3/2}\,dx$, (b) $\displaystyle\int_1^\infty x^{-1/2}\,dx$, (c) $\displaystyle\int_0^1 x^{-3/2}\,dx$, (d) $\displaystyle\int_0^1 x^{-1/2}\,dx$.

17.18 ⭐⭐ Rewrite each integrand as a single power $x^{-p}$, then classify: (a) $\displaystyle\int_2^\infty \frac{1}{x\sqrt{x}}\,dx$, (b) $\displaystyle\int_0^1 \frac{1}{x\sqrt{x}}\,dx$, (c) $\displaystyle\int_1^\infty \frac{1}{\sqrt[4]{x^5}}\,dx$.

17.19 ⭐⭐ A single integral can be improper at both ends. Split $\displaystyle\int_0^\infty x^{-p}\,dx$ at $x=1$ and explain why it diverges for every real $p$. (Use the two p-rules on the two pieces.)

17.20 ⭐⭐⭐ For which exponents $p$ does $\displaystyle\int_0^1 \frac{1}{x^p\,(\,-\ln x\,)}\,dx$ converge? (Hint: substitute $u=-\ln x$ and use the comparison test of Section 17.3.)

Part D — Comparison and Limit Comparison (Section 17.3)

17.21 ⭐⭐ Use direct comparison to show $\displaystyle\int_1^\infty \frac{1}{1+x^3}\,dx$ converges (compare with $1/x^3$).

17.22 ⭐⭐ Use direct comparison to show $\displaystyle\int_1^\infty \frac{\sin^2 x}{x^2}\,dx$ converges. (Hint: $\sin^2 x \le 1$.)

17.23 ⭐⭐ Show $\displaystyle\int_1^\infty \frac{e^x}{x}\,dx$ diverges by comparison with $1/x$ (compare Worked Example 17.3.2).

17.24 ⭐⭐⭐ Use limit comparison to determine convergence of $\displaystyle\int_1^\infty \frac{2x+1}{x^4+2x}\,dx$. Choose the comparison power by keeping only leading terms.

17.25 ⭐⭐⭐ Determine convergence of $\displaystyle\int_2^\infty \frac{1}{\sqrt{x^4-1}}\,dx$. (Hint: the dominant tail behavior is $1/x^2$.)

17.26 ⭐⭐⭐ Show $\displaystyle\int_0^1 \frac{\sin x}{x^{3/2}}\,dx$ converges. (Hint: near $0$, $\sin x \approx x$, so the integrand behaves like $x^{-1/2}$; this is a Type 2 comparison.)

17.27 ⭐⭐⭐ The integrand $\dfrac{\sin x}{x}$ changes sign, so the comparison tests of Section 17.3 do not apply to it directly. Explain precisely why, and state what the tests can establish about $\int_1^\infty \frac{\sin x}{x}\,dx$ if you test $\left|\frac{\sin x}{x}\right|$ instead (Section 17.7).

Part E — The Gamma and Gaussian Integrals (Sections 17.4, 17.5)

17.28 ⭐⭐ Compute $\Gamma(5) = \displaystyle\int_0^\infty x^4 e^{-x}\,dx$ using $\Gamma(n) = (n-1)!$. State the value.

17.29 ⭐⭐ Using $\Gamma(s+1) = s\,\Gamma(s)$ and $\Gamma(1/2)=\sqrt{\pi}$, compute $\Gamma(3/2)$, $\Gamma(5/2)$, and $\Gamma(7/2)$.

17.30 ⭐⭐⭐ Express $\displaystyle\int_0^\infty x^{1/2}\,e^{-x^2}\,dx$ in terms of a Gamma value via the substitution $u = x^2$, then evaluate it as a multiple of $\sqrt{\pi}$ if possible.

17.31 ⭐⭐⭐ Starting from $\displaystyle\int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}$ (Section 17.5), show by the substitution $u = x/\sqrt{2}$ that $\displaystyle\int_{-\infty}^\infty e^{-x^2/2}\,dx = \sqrt{2\pi}$, and explain why this is exactly the $1/\sqrt{2\pi}$ in the normal density.

17.32 ⭐⭐⭐⭐ Prove the half-integer formula $\displaystyle\Gamma\!\left(n+\tfrac12\right) = \frac{(2n)!}{4^n\,n!}\sqrt{\pi}$ for non-negative integers $n$ by induction, using $\Gamma(1/2)=\sqrt{\pi}$ and the recursion $\Gamma(s+1)=s\,\Gamma(s)$.

Part F — Applications (Sections 17.4, 17.5, 17.6) ⭐⭐⭐⭐

17.33 — Pharmacokinetics (biology / medicine). A drug's blood concentration after a bolus dose is $C(t) = C_0 e^{-kt}$. (a) Verify the area-under-the-curve exposure $\text{AUC} = \int_0^\infty C_0 e^{-kt}\,dt = C_0/k$ (Section 17.6). (b) Compute the mean residence time $\dfrac{\int_0^\infty t\,C(t)\,dt}{\int_0^\infty C(t)\,dt}$ and show it equals $1/k$. (Recognize a Gamma integral in the numerator.)

17.34 — Present value of a perpetuity (economics). An asset pays a continuous income stream at constant rate $R$ dollars per year forever, discounted at continuous rate $r>0$. Its present value is $\text{PV} = \int_0^\infty R\,e^{-rt}\,dt$. (a) Evaluate the integral and show $\text{PV} = R/r$. (b) Explain why the integral diverges if $r = 0$, and interpret that financially.

17.35 — Escape velocity (physics). The work to lift a mass $m$ from radius $r_0$ to infinity against gravity is $W = \int_{r_0}^\infty \frac{GMm}{r^2}\,dr$. (a) Evaluate it and show $W = GMm/r_0$. (b) Set $\tfrac12 m v_e^2 = W$ and solve for the escape velocity $v_e = \sqrt{2GM/r_0}$. (c) Explain, using the p-integral rule of Section 17.1, why a hypothetical $1/r$ force law would make escape impossible at any finite speed.

17.36 — Normalization of a probability density (data science / statistics). The exponential density is $f(x) = \lambda e^{-\lambda x}$ for $x\ge 0$. (a) Verify $\int_0^\infty f(x)\,dx = 1$. (b) Compute the mean $E[X] = \int_0^\infty x\,f(x)\,dx$ and show it equals $1/\lambda$. (c) Compute $E[X^2]$ and use it to find the variance $1/\lambda^2$. (Both moments are Gamma integrals.)

Part G — Theoretical and Open-Ended (⭐⭐⭐⭐)

17.37 ⭐⭐⭐⭐ For $p$ slightly greater than $1$, the integral $\int_1^\infty x^{-p}\,dx = \frac{1}{p-1}$ converges but its value explodes. (a) Confirm the formula. (b) Compute $\lim_{p\to 1^+}\frac{1}{p-1}$ and connect this blow-up to the divergence of $\int_1^\infty \frac{1}{x}\,dx$. Discuss what this says about the borderline $p=1$.

17.38 ⭐⭐⭐⭐ (Open-ended, modeling.) Find a quantity from your chosen portfolio track — biology, economics, physics, or data science — that is naturally expressed as an improper integral over an unbounded domain or across a singularity. Set it up, determine convergence using the tools of this chapter, evaluate it if possible, and write a paragraph interpreting the result. Connect it explicitly to the integral test foreshadowed for Chapter 22.

Solutions

Selected solutions appear in appendices/answers-to-selected.md.

Tier Summary

Total: 38 problems — roughly 8–13 hours of focused practice.