Calculus 2 problem set: integration techniques, the Gaussian, series, Taylor, polar, ODEs

Overview

Module	S-M02 · solve · none · Pass 2 · 6 to 8 h
You build	answers in solve/S-M02.toml (48 checked by SymPy) and 4 proofs in solve/S-M02/q12.md, q18.md, q30.md, q40.md (self-graded against their rubrics)
Contract	none: a pen and paper set
Tests	course/solve/S-M02/key.toml (hidden): typed answers plus reject canaries; the problems are in course/solve/S-M02/problems.md and in section 4
Needs	no module. Reading: S-M01 (derivatives and the fundamental theorem) and the Calculus 2 topic
Used by	no call site (a solve set). Take it after M02.1 (Taylor series in code) and M02.2 (the EMA as a geometric series) in Pass 2; M07.0 and M07.3 read its Gaussian integrals, which are the solve-only M02.3
Milestone	MS-P2 (the Pass 2 gate runs ol check on every solve part of the pass)
Optional depth	OpenStax, Calculus Volume 2 (free), ch. 3 to 7; Trefethen, Approximation Theory and Approximation Practice, ch. 1 to 3, for why a few Taylor terms after range reduction are enough

Key Takeaways

• Integration by parts is the product rule integrated; substitution is the chain rule integrated (q1 to q11, q12).
• The Gaussian integral $\int e^{-x^2} = \sqrt{\pi}$ gives the normal density its constant, and half the second moment of a standard normal sits on each side of 0, which is the ReLU factor in Kaiming initialization (q16, q17).
• A series converges when its partial sums do; terms going to 0 is necessary, not sufficient (q20, q30).
• A Taylor polynomial plus a Lagrange remainder is an approximation with a guarantee; range reduction keeps the remainder small (q35, q38).
• A first-order ODE describes a rate; the logistic equation’s solution is the sigmoid, and Euler’s method is one step of gradient descent on gradient flow (q48, q51, q52).

How to work this chapter

ol start S-M02             # writes solve/S-M02.toml and the four proof files
ol check S-M02             # SymPy checks the answers, then asks each proof rubric (y/n)
ol check S-M02 --regrade   # ask the rubrics again after you change a proof

---

1. Why now

Your Pass 2 code leans on this calculus in three places. M02.1 computes $e^x$ and $\operatorname{erf}$ from Taylor polynomials, and its tests check a Lagrange remainder bound; M09.6 later puts the same polynomial in C. M02.2 treats the exponential moving average as a geometric series and derives Adam’s bias correction (M10.3) from its partial sum. And M07.0 and M07.3 draw normal random numbers and pick initialization scales from integrals of the normal density, which only exist because $\int e^{-x^2}$ converges to $\sqrt{\pi}$. This set checks the hand techniques behind those modules: integrating by parts and by substitution, improper integrals, convergence, Taylor remainders, curves, and the simplest differential equations.

2. Principles

Symbol	Meaning	Type / shape
$u, v$	functions in integration by parts	functions
$\int_a^\infty f$	improper integral, $\lim_{b \to \infty} \int_a^b f$	real or divergent
$\varphi(z)$	standard normal density, $e^{-z^2/2}/\sqrt{2\pi}$	function
$\sum_{n} a_n$, $S_N$	a series and its partial sum $S_N = \sum_{n \le N} a_n$	real
$R$	radius of convergence of a power series $\sum c_n x^n$	nonnegative real
$T_n(x)$	degree $n$ Taylor polynomial at 0, $\sum_{k \le n} f^{(k)}(0)\, x^k / k!$	polynomial
$R_n(x)$	remainder $f(x) - T_n(x)$	real
$r, \theta$	polar coordinates: $x = r\cos\theta$, $y = r\sin\theta$	reals
$y(t)$, $y'$	an unknown function of time and its derivative	function
$h$	the step size of Euler’s method	positive real

2.1 Integration techniques

Substitution reverses the chain rule: $\int f(g(x))\,g'(x)\,dx = \int f(u)\,du$ with $u = g(x)$; change the limits with it. Integration by parts reverses the product rule: $\int_a^b u v' = [uv]_a^b - \int_a^b u' v$; pick $u$ to get simpler when differentiated ($x$, $\ln x$). Partial fractions split a rational function into simple pieces: $\frac{1}{x^2 - 1} = \frac{1}{2}\left(\frac{1}{x - 1} - \frac{1}{x + 1}\right)$. Trigonometric identities such as $\sin^2 x = \frac{1 - \cos 2x}{2}$ and substitutions such as $x = 2\sin\theta$ remove square roots.

2.2 Improper integrals and the Gaussian

An integral over an infinite range, or of a function that blows up at an endpoint, is defined as a limit, and it converges when the limit is finite: $\int_1^\infty x^{-2} = 1$ but $\int_1^\infty x^{-1} = \infty$; $\int_0^1 x^{-1/2} = 2$ although the integrand is unbounded. The Gaussian integral $I = \int_{-\infty}^\infty e^{-x^2} dx$ has no elementary antiderivative, but $I^2$ is a double integral over the plane, and in polar coordinates ($dx\,dy = r\,dr\,d\theta$) it becomes $\int_0^{2\pi}\int_0^\infty e^{-r^2} r\,dr\,d\theta = \pi$. Substituting $x = z/\sqrt{2}$ gives $\int e^{-z^2/2} dz = \sqrt{2\pi}$, the constant of the normal density. By symmetry, $\int_0^\infty z^2 \varphi(z)\,dz$ is half of $\int_{-\infty}^\infty z^2 \varphi(z)\,dz = 1$.

2.3 Series and convergence

A series $\sum a_n$ converges to $S$ when its partial sums $S_N \to S$. Tests: a geometric series $\sum_{n \ge 0} r^n = \frac{1}{1 - r}$ for $|r| < 1$; the $p$-series $\sum n^{-p}$ converges exactly for $p > 1$ (compare with $\int_1^\infty x^{-p}$, the integral test); the ratio test gives convergence when $|a_{n+1}/a_n| \to L < 1$; an alternating series with terms decreasing to 0 converges. A power series $\sum c_n x^n$ converges for $|x| < R$ and diverges for $|x| > R$; each endpoint $x = \pm R$ needs its own test.

2.4 Taylor polynomials and the remainder

$T_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(0)}{k!} x^k$ matches $f$ and its first $n$ derivatives at 0. Lagrange’s form of the remainder says $f(x) - T_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!} x^{n+1}$ for some $c$ between 0 and $x$, so a bound on $f^{(n+1)}$ bounds the error. The error grows like $|x|^{n+1}$, so implementations shrink $x$ first: range reduction writes $x = k\ln 2 + r$ with $|r| \le \frac{1}{2}\ln 2$ and computes $e^x = 2^k e^r$, where a degree 6 polynomial in $r$ already reaches float32 precision (M02.1, M09.6).

2.5 Parametric curves and polar coordinates

A curve $x(t), y(t)$ has slope $\frac{dy}{dx} = \frac{y'(t)}{x'(t)}$ and arc length $\int \sqrt{x'(t)^2 + y'(t)^2}\,dt$. In polar coordinates the area swept by $r(\theta)$ is $\int \frac{1}{2} r^2\,d\theta$, and multiplying an equation by $r$ converts it with $r^2 = x^2 + y^2$, $r\cos\theta = x$, $r\sin\theta = y$.

2.6 First-order differential equations

$y' = -k y$ has the solution $y(t) = y(0)\,e^{-kt}$ (separate variables: $dy/y = -k\,dt$). A linear equation $y' + p\,y = q(t)$ is solved with the integrating factor $e^{\int p}$. The logistic equation $y' = y(1 - y)$ separates by partial fractions and gives the sigmoid. Euler’s method steps $y_{k+1} = y_k + h\,f(t_k, y_k)$. Gradient descent with learning rate $h$ is exactly Euler’s method on the gradient flow $x' = -\nabla f(x)$, and M02.4 shows momentum is Euler on the heavy-ball equation.

3. Worked example by hand

This is a sibling of q1 and q17, not one of the graded problems.

By parts, twice. Compute $\int_0^1 x^2 e^{-x}\,dx$. Take $u = x^2$, $v' = e^{-x}$, so $u' = 2x$, $v = -e^{-x}$: the integral is $[-x^2 e^{-x}]_0^1 + 2\int_0^1 x e^{-x}\,dx = -e^{-1} + 2\int_0^1 x e^{-x}\,dx$. Again with $u = x$: $\int_0^1 x e^{-x} = [-x e^{-x}]_0^1 + \int_0^1 e^{-x} = -e^{-1} + (1 - e^{-1}) = 1 - 2e^{-1}$. Total: $-e^{-1} + 2 - 4e^{-1} = 2 - 5e^{-1} \approx 0.1606$. In solve/ this is answer = "2 - 5*exp(-1)"; answer = "0.1606" fails as inexact.

A Gaussian moment. $\mathbb{E}[|Z|] = 2\int_0^\infty z\,\varphi(z)\,dz$ for a standard normal $Z$. Substitute $u = z^2/2$, $du = z\,dz$: $\int_0^\infty z e^{-z^2/2} dz = \int_0^\infty e^{-u} du = 1$. So $\mathbb{E}[|Z|] = \frac{2}{\sqrt{2\pi}} = \sqrt{2/\pi} \approx 0.798$.

4. The problem set

Write each answer in solve/S-M02.toml:

[q3]
answer = "pi/2 - 1"
[q19]
answer = "true"
[q28]
answer = "[-1, 1)"
[q45]
answer = "x^2 + y^2 = 2*y"
[q12]
proof = "S-M02/q12.md"

Numbers are exact (log(3/2)/2, not 0.2027). Taylor polynomials are checked symbolically, in any term order. Write $e$ as E or exp(1) and $\pi$ as pi.

Integration techniques

q1. $\displaystyle\int_0^1 x e^x\, dx$ (by parts). [number]

q2. $\displaystyle\int_1^e \ln x\, dx$. [number]

q3. $\displaystyle\int_0^{\pi/2} x \cos x\, dx$. [number]

q4. $\displaystyle\int_0^1 \frac{x}{1 + x^2}\, dx$ (substitution). [number]

q5. $\displaystyle\int_0^{\pi/2} \sin^2 x\, dx$. [number]

q6. $\displaystyle\int_2^3 \frac{1}{x^2 - 1}\, dx$ (partial fractions). [number]

q7. Give the antiderivative $F$ of $x^2 e^x$ with $F(0) = 0$. [expr in x]

q8. $\displaystyle\int_0^1 x \sqrt{1 - x^2}\, dx$. [number]

q9. $\displaystyle\int_0^\pi x \sin x\, dx$. [number]

q10. $\displaystyle\int_0^1 x^3 e^{x^2}\, dx$. [number]

q11. $\displaystyle\int_0^1 \frac{1}{\sqrt{4 - x^2}}\, dx$ (substitute $x = 2\sin\theta$). [number]

q12. Derive the integration by parts formula $\int_a^b u\,v'\,dx = \big[u v\big]_a^b - \int_a^b u'\,v\,dx$ from the product rule and the fundamental theorem of calculus. [proof]

Improper integrals and the Gaussian

q13. $\displaystyle\int_1^\infty \frac{1}{x^2}\, dx$. [number]

q14. $\displaystyle\int_0^\infty e^{-2x}\, dx$. [number]

q15. $\displaystyle\int_0^1 \frac{1}{\sqrt{x}}\, dx$ (improper at 0). [number]

q16. $\displaystyle\int_{-\infty}^{\infty} e^{-x^2}\, dx$. [number]

q17. $Z$ is a standard normal with density $\varphi(z) = e^{-z^2/2}/\sqrt{2\pi}$. Give $\mathbb{E}[\max(Z, 0)^2] = \displaystyle\int_0^\infty z^2 \varphi(z)\, dz$, the second moment of a ReLU of a standard normal. [number]

q18. Prove $\displaystyle\int_{-\infty}^{\infty} e^{-x^2}\, dx = \sqrt{\pi}$ by squaring the integral and changing to polar coordinates. [proof]

Series and convergence tests

q19. Does $\displaystyle\sum_{n=1}^\infty \frac{1}{n^2}$ converge? [bool]

q20. Does $\displaystyle\sum_{n=1}^\infty \frac{1}{n}$ converge? [bool]

q21. $\displaystyle\sum_{n=0}^\infty \left(\frac{2}{3}\right)^n$. [number]

q22. $\displaystyle\sum_{n=1}^\infty \frac{1}{n(n + 1)}$ (telescoping). [number]

q23. $\displaystyle\sum_{n=1}^\infty \frac{n}{2^n}$. [number]

q24. Does $\displaystyle\sum_{n=1}^\infty \frac{n!}{n^n}$ converge? (Ratio test.) [bool]

q25. Does the alternating series $\displaystyle\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}$ converge? [bool]

q26. Give the sum $\displaystyle\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}$. [number]

q27. Give the radius of convergence of $\displaystyle\sum_{n=1}^\infty \frac{x^n}{n}$. [number]

q28. Give the interval of convergence of $\displaystyle\sum_{n=1}^\infty \frac{x^n}{n}$, endpoints included or not. [interval]

q29. Give the set of real $p$ for which $\displaystyle\sum_{n=1}^\infty \frac{1}{n^p}$ converges. [interval]

q30. Prove that $\displaystyle\sum_{n=1}^\infty \frac{1}{n}$ diverges. [proof]

Taylor series and remainders

q31. Give the degree 3 Taylor polynomial of $e^x$ at 0. [expr in x]

q32. Give the Maclaurin series of $\ln(1 + x)$ through the $x^3$ term. [expr in x]

q33. Give the coefficient of $x^5$ in the Maclaurin series of $\sin x$. [number]

q34. Give the degree 4 Taylor polynomial of $\cos x$ at 0. [expr in x]

q35. With the Lagrange remainder, bound $|e^x - T_3(x)|$ for $|x| \le 1/2$, where $T_3$ is the degree 3 Taylor polynomial at 0. Use $e^c \le e^{1/2}$ for the unknown point $c$ and give the bound. [number]

q36. Give the smallest $n$ with $\dfrac{1}{(n + 1)!} < 10^{-6}$. [number]

q37. Give the degree 3 Taylor polynomial of $\dfrac{1}{1 - x}$ at 0. [expr in x]

q38. Range reduction writes $x = k \ln 2 + r$ with $k$ the integer nearest to $x / \ln 2$, so that $e^x = 2^k e^r$ with $|r| \le \frac{1}{2}\ln 2$. For $x = 5$, give $r$. [number]

q39. $\operatorname{erf}(x) = \dfrac{2}{\sqrt{\pi}} \displaystyle\int_0^x e^{-t^2}\, dt$. Give the first three nonzero terms of its Maclaurin series. [expr in x]

q40. Prove that $\left|\sin x - \left(x - \dfrac{x^3}{6}\right)\right| \le \dfrac{|x|^5}{120}$ for every real $x$. [proof]

Parametric curves and polar coordinates

q41. The curve $x = t^2$, $y = t^3$. Give $dy/dx$ at $t = 1$. [number]

q42. Give the arc length of $x = \cos t$, $y = \sin t$ for $0 \le t \le \pi/2$. [number]

q43. Give the arc length of $x = t^2$, $y = t^3$ for $0 \le t \le 1$. [number]

q44. Give the area enclosed by the polar curve $r = 2\cos\theta$, $-\pi/2 \le \theta \le \pi/2$. [number]

q45. Rewrite the polar curve $r = 2\sin\theta$ as an equation in $x$ and $y$. [equation in x, y]

q46. Give the area enclosed by the cardioid $r = 1 + \cos\theta$, $0 \le \theta \le 2\pi$. [number]

First-order differential equations

q47. Solve $y' = -2y$ with $y(0) = 3$. [expr in t]

q48. Solve the logistic equation $y' = y(1 - y)$ with $y(0) = 1/2$. [expr in t]

q49. Solve $y' + y = t$ with $y(0) = 0$. [expr in t]

q50. A quantity decays by $y' = -k y$ and halves every 10 time units. Give $k$. [number]

q51. Euler’s method on $y' = y$, $y(0) = 1$, with step $h = 1/2$: give the approximation of $y(1)$ after two steps. [number]

q52. Gradient flow on $f(x) = x^2/2$ is $x'(t) = -f'(x) = -x$. Starting from $x(0) = 4$, give the time at which $x(t) = 1$. [number]

5. Pitfalls

Pitfall	Symptom	Caught by
Losing the constant from $du = g'(x)\,dx$	substitution answers off by a factor of 2	q4 (canary log(2)), q6 (canary log(3/2)), q10 (canary 1)
Treating an unbounded integrand as a divergent integral	$\int_0^1 x^{-1/2}$ reported as infinite	q15 (canary oo)
Confusing the normalizers of $e^{-x^2}$ and $e^{-x^2/2}$	normal densities off by $\sqrt{2}$	q16 (canary sqrt(2*pi))
Concluding convergence from terms that go to 0	the harmonic series called convergent	q20, q30 (proof)
Testing only the interior of an interval of convergence	an endpoint included or dropped wrongly	q28 (canaries), q29 (canary [1, oo))
Using the remainder of the wrong degree, or dropping its derivative factor	an error bound that is too optimistic	q35 (canaries)
Rounding the range-reduction quotient the wrong way	$	r
Ignoring the initial condition of an ODE	a family of solutions instead of one	q47, q49 (canaries)

6. Where it’s used next

Direction	Module	How it uses this
Back	S-M01	derivatives, antiderivatives, and the fundamental theorem
Forward	M02.1	Taylor polynomials with range reduction for exp and erf, tested against the Lagrange bound of q35
Forward	M02.2	the EMA as a truncated geometric series (q21) and its bias correction
Forward	M07.0	the normal density’s constant (q16) behind Box-Muller normals
Forward	M07.3	the ReLU second moment of q17 sets Kaiming’s gain $\sqrt{2}$
Forward	M02.4	Euler’s method and gradient flow (q51, q52) explain momentum
Forward	S-M04	multiple integrals and the change of variables behind q18