# UW-Madison MATH 222 Calculus 2: Topics, Study Guide and Practice

> A study guide for MATH 222 at UW-Madison: integration techniques, improper integrals, differential equations, Taylor series and vectors, with practice.

Source: https://practicecalculus.com/courses/uw-madison-math-222
Updated: 2026-11-03

## At a glance

- **Course code:** MATH 222
- **Credits:** 4
- **Requisites:** MATH 217 or MATH 221 (MATH 211 and 213 do not count)
- **Covers:** Techniques of integration, improper integrals, first-order ordinary differential equations, sequences and series, Taylor series, vector geometry in two and three dimensions
- **Previous course:** MATH 221 (Calculus and Analytic Geometry 1)
- **Next course:** MATH 234 (Calculus: Functions of Several Variables)
- **Practice here:** [Calculus 2](/learn/calculus-2) plus the vectors unit of [Calculus 3](/learn/calculus-3)

## Key takeaways

- MATH 222 is the 4-credit second semester of Calculus and Analytic Geometry at UW-Madison.
- It covers integration techniques, improper integrals, first-order differential equations, sequences, series and Taylor series.
- It ends with vector geometry in two and three dimensions, which prepares you for MATH 234.
- You need MATH 221 or MATH 217; the survey courses MATH 211 and 213 do not meet the requisite.

**MATH 222** is the second semester of Calculus and Analytic Geometry at the University of Wisconsin-Madison. Students take it after MATH 221 on the way to MATH 234, and it is required for most engineering, physics and math programs. This guide maps every MATH 222 topic to free practice, so you can learn each technique by working problems with instant feedback.

## What MATH 222 covers

The UW-Madison Guide lists six themes for MATH 222: techniques of integration, improper integrals, first-order ordinary differential equations, sequences and series, Taylor series, and vector geometry in two and three dimensions.

The first five match a standard **Calculus 2** course. The vector geometry at the end (vectors, dot and cross products, lines and planes) is material that many schools teach at the start of **Calculus 3**, so MATH 234 can move straight into functions of several variables.

## MATH 222 topic map

| MATH 222 topic | Practice | Read |
| --- | --- | --- |
| Integration by parts | [Integration by parts](/learn/calculus-2/integration-techniques/integration-by-parts) | [Integration by parts](/blog/integration-by-parts) |
| Trig integrals and trig substitution | [Trig substitution](/learn/calculus-2/integration-techniques/trig-substitution) | [Trig substitution](/blog/trig-substitution) |
| Partial fractions | [Partial fractions](/learn/calculus-2/integration-techniques/partial-fractions) | [Partial fractions](/blog/partial-fractions) |
| Improper integrals | [Integrating to infinity](/learn/calculus-2/improper-integrals-and-applications/infinite-limits-of-integration) | [Improper integrals](/blog/improper-integrals) |
| First-order differential equations | [Separable differential equations](/learn/calculus-2/improper-integrals-and-applications/separable-equations) | [Exponential functions](/functions/exponential-functions) |
| Sequences | [Sequences and their limits](/learn/calculus-2/sequences-and-series/sequences) | [Limits at infinity](/blog/limits-at-infinity) |
| Series and convergence tests | [Choosing a test](/learn/calculus-2/sequences-and-series/choosing-a-test) | [Convergence tests](/blog/convergence-tests), [ratio test](/rules/ratio-test) |
| Geometric series | [Geometric series](/learn/calculus-2/sequences-and-series/geometric-series) | [Geometric series](/blog/geometric-series) |
| Power and Taylor series | [Common Maclaurin series](/learn/calculus-2/power-series-parametric-polar/maclaurin-series) | [Taylor series](/blog/taylor-series), [power series](/blog/power-series) |
| Vectors, dot and cross product | [The cross product](/learn/calculus-3/vectors-and-space/cross-product) | [Vectors, dot and cross product](/blog/vectors-dot-and-cross-product) |
| Lines and planes in space | [Planes in space](/learn/calculus-3/vectors-and-space/planes-in-space) | [Lines in space](/learn/calculus-3/vectors-and-space/lines-in-space) |

## MATH 222 in the sequence

| Course | Focus | Requisite |
| --- | --- | --- |
| MATH 221 | Derivatives, integrals, analytic geometry | MATH 114, or 112 and 113, or placement |
| MATH 222 | Integration techniques, ODEs, series, vectors | MATH 217 or MATH 221 |
| MATH 234 | Calculus of several variables and vector calculus | MATH 222 |
| MATH 213 | Survey of Calculus 2 (business and social science) | MATH 211, 217 or 221 |

MATH 213 also touches series and differential equations, but it is a survey course aimed at business and social science applications and does not lead to MATH 234. For the full path, see the [MATH 221 guide](/courses/uw-madison-math-221) and the [MATH 234 guide](/courses/uw-madison-math-234).

## Example problems at the MATH 222 level

**1. A separable differential equation.** Solve $\frac{dy}{dx} = 2xy$ with $y(0) = 3$. Separate: $\frac{dy}{y} = 2x\,dx$, so $\ln|y| = x^2 + C$ and $y = Ae^{x^2}$. The initial condition gives $A = 3$:

$$y = 3e^{x^2}$$

**2. A series test.** Does $\sum_{n=1}^{\infty} \frac{n}{3^n}$ converge? Use the ratio test:

$$\lim_{n \to \infty} \frac{n+1}{3^{n+1}} \cdot \frac{3^n}{n} = \lim_{n \to \infty} \frac{n+1}{3n} = \frac{1}{3} < 1$$

so the series converges.

**3. A cross product.** For $\mathbf{u} = \langle 1, 0, 0 \rangle$ and $\mathbf{v} = \langle 0, 1, 1 \rangle$, $\mathbf{u} \times \mathbf{v} = \langle 0 \cdot 1 - 0 \cdot 1,\ 0 \cdot 0 - 1 \cdot 1,\ 1 \cdot 1 - 0 \cdot 0 \rangle = \langle 0, -1, 1 \rangle$, which is perpendicular to both.

## How to do well in MATH 222

- **Build an integration decision habit.** Before integrating, ask: substitution, parts, trig, or partial fractions? The [choosing a strategy](/learn/calculus-2/integration-techniques/choosing-a-strategy) lesson drills that first step.
- **Treat series as a checklist.** Try the [nth-term test](/rules/nth-term-test) first, then compare to a geometric or p-series, then reach for the ratio test. Practice the order until it is automatic.
- **Memorize five Maclaurin series.** $e^x$, $\sin x$, $\cos x$, $\frac{1}{1-x}$ and $\ln(1+x)$ answer a surprising share of Taylor questions. Keep them on your [formula sheet](/formulas).
- **Review the usual traps** in [common calculus mistakes](/blog/common-calculus-mistakes), especially forgetting $+C$ and losing signs in parts.
- **Practice under exam conditions** with mixed sets in [practice](/practice), using the plan in [how to prepare for a calculus exam](/blog/how-to-prepare-for-a-calculus-exam).

> **Tip:** the [geometric series formula](/rules/geometric-series-formula) is the backbone of MATH 222 series. Power series, comparison tests and Taylor series all lean on it.

## Frequently asked questions

### What is MATH 222 at UW-Madison?

MATH 222, Calculus and Analytic Geometry 2, is the second course in the main UW-Madison calculus sequence. It covers integration techniques, improper integrals, first-order ODEs, sequences and series, Taylor series and vector geometry.

### What do I need before MATH 222?

The Guide lists MATH 217 or MATH 221 as the requisite. MATH 211 and MATH 213 do not fulfill it.

### What comes after MATH 222?

MATH 234, Calculus: Functions of Several Variables, which requires MATH 222.

### Is MATH 222 harder than MATH 221?

Many students find it harder, mainly because of series and because integration requires choosing a technique. The skills are very learnable with steady practice, especially if derivatives and substitution are already automatic.

### How do I get ready for series in MATH 222?

Start with geometric series and the nth-term test, then learn when to use each convergence test. The sequences and series unit on PracticeCalculus is built around choosing the right test.