# UCLA Math 31B Integration and Infinite Series: Study Guide

> A study guide for UCLA Math 31B: the definite integral, integration techniques, convergence tests, power series and Taylor series, with free practice.

Source: https://practicecalculus.com/courses/ucla-math-31b
Updated: 2026-11-03

## At a glance

- **Course code:** MATH 31B
- **Units:** 4 (three hours of lecture and one of discussion per week)
- **Requisite:** MATH 31A with a grade of C- or better
- **Covers:** Transcendental functions, the definite integral and the Fundamental Theorem, substitution, integration by parts, trig integrals and substitution, partial fractions, sequences, series and convergence tests, power series and Taylor series
- **Credit restriction:** Not open for credit to students with credit for MATH 3B
- **Next course:** MATH 32B (Calculus of Several Variables), which requires both MATH 31B and MATH 32A
- **Practice here:** [Calculus 2](/learn/calculus-2), plus the integrals unit of [Calculus 1](/learn/calculus-1)

## Key takeaways

- Math 31B is UCLA's second calculus course: transcendental functions, methods and applications of integration, and sequences and series.
- In the department outline used from Summer 2026, it opens with the definite integral and the Fundamental Theorem of Calculus.
- The final third of the course is infinite series, ending with power series and Taylor series.
- It requires Math 31A with a C- or better and, together with Math 32A, unlocks Math 32B.

**UCLA Math 31B** is the second quarter of UCLA's calculus sequence, after [Math 31A](/courses/ucla-math-31a). It completes single-variable calculus: the definite integral and its techniques, then sequences and series. Along with [Math 32A](/courses/ucla-math-32a), it is required before Math 32B. This guide maps every Math 31B topic to free practice with instant feedback.

## What UCLA Math 31B covers

The UCLA General Catalog lists transcendental functions, methods and applications of integration, and sequences and series. The Mathematics Department's outline for Summer 2026 onward organizes that into three units. The first is the definite integral: a review of antiderivatives and separable equations, Riemann sums, both parts of the Fundamental Theorem, net change, substitution, integrals of exponential, log and inverse trig functions, and area and volume. The second is methods of integration: integration by parts, trig integrals, trig substitution, partial fractions, choosing a strategy, and first-order linear differential equations. The third is infinite series: sequences, the divergence, integral, p-series and comparison tests, absolute and conditional convergence, the ratio and root tests, power series and Taylor series.

Compared with a semester sequence, Math 31B covers the integration end of **Calculus 1** and most of **Calculus 2**.

## Math 31B topic map

| Math 31B topic | Practice | Read |
| --- | --- | --- |
| Riemann sums and the definite integral | [Area with rectangles](/learn/calculus-1/integrals/riemann-sums) | [Riemann sums guide](/blog/riemann-sums) |
| Fundamental Theorem of Calculus | [Derivatives of integrals](/learn/calculus-1/integrals/fundamental-theorem-part-1) | [FTC guide](/blog/fundamental-theorem-of-calculus), [FTC part 2](/theorems/fundamental-theorem-of-calculus-part-2) |
| Net change | [Net change and displacement](/learn/calculus-1/applications-of-integrals/net-change) | [Definite integral properties](/rules/definite-integral-properties) |
| Substitution | [U-substitution](/learn/calculus-1/integrals/u-substitution) | [U-substitution guide](/blog/u-substitution) |
| Area and volume | [Volumes with disks](/learn/calculus-1/applications-of-integrals/volumes-disks) | [Area between curves](/blog/area-between-curves), [volumes of revolution](/blog/volumes-of-revolution) |
| Integration by parts | [Integration by parts](/learn/calculus-2/integration-techniques/integration-by-parts) | [Integration by parts guide](/blog/integration-by-parts) |
| Trig integrals and trig substitution | [Powers of sine and cosine](/learn/calculus-2/integration-techniques/trig-integrals) | [Trig substitution guide](/blog/trig-substitution) |
| Partial fractions | [Partial fractions](/learn/calculus-2/integration-techniques/partial-fractions) | [Partial fractions guide](/blog/partial-fractions) |
| Strategy for integration | [Choosing a strategy](/learn/calculus-2/integration-techniques/choosing-a-strategy) | [Integration by parts rule](/rules/integration-by-parts-rule) |
| Differential equations | [Separable differential equations](/learn/calculus-2/improper-integrals-and-applications/separable-equations) | [Natural exponential function](/functions/natural-exponential-function) |
| Sequences | [Sequences and their limits](/learn/calculus-2/sequences-and-series/sequences) | [Limits at infinity](/blog/limits-at-infinity) |
| Divergence, integral and comparison tests | [The integral test and p-series](/learn/calculus-2/sequences-and-series/integral-test-and-p-series) | [Convergence tests](/blog/convergence-tests), [nth-term test](/rules/nth-term-test) |
| Ratio and root tests | [Ratio and root tests](/learn/calculus-2/sequences-and-series/ratio-and-root-tests) | [Ratio test](/rules/ratio-test) |
| Power series and Taylor series | [Common Maclaurin series](/learn/calculus-2/power-series-parametric-polar/maclaurin-series) | [Power series](/blog/power-series), [Taylor series](/blog/taylor-series) |

## Math 31B vs Math 3B vs Math 32A

| Course | Requisite | Content |
| --- | --- | --- |
| Math 31B | 31A | Integration techniques and infinite series |
| Math 3B | 3A | Calculus for life sciences: integration, differential equations, linear models in biology |
| Math 32A | 31A | Vectors and differential calculus of several variables |

Math 31B and Math 3B exclude credit for each other. Math 32A can come before, after or alongside 31B, but [Math 32B](/courses/ucla-math-32b) needs both.

## Example problems at the Math 31B level

**1. Integration by parts with an exponential.** With $u = x$, $dv = e^{2x}\,dx$:

$$\int x e^{2x}\,dx = \frac{x e^{2x}}{2} - \int \frac{e^{2x}}{2}\,dx = \frac{x e^{2x}}{2} - \frac{e^{2x}}{4} + C$$

**2. An inverse trig integral.**

$$\int_0^1 \frac{dx}{1 + x^2} = \arctan 1 - \arctan 0 = \frac{\pi}{4}$$

**3. The limit comparison test.** For $\sum \frac{1}{n^2 + n}$, compare with $\sum \frac{1}{n^2}$:

$$\lim_{n\to\infty}\frac{1/(n^2 + n)}{1/n^2} = \lim_{n\to\infty}\frac{n^2}{n^2 + n} = 1$$

Since $\sum \frac{1}{n^2}$ is a convergent p-series, the original series converges.

## How to do well in UCLA Math 31B

- **Drill technique recognition.** Exams mix integrals without telling you the method. The [techniques practice](/learn/calculus-2/integration-techniques/techniques-practice) set builds that instinct.
- **Organize the series tests.** Learn when to use each, starting with the [nth-term test](/rules/nth-term-test) and the [geometric series formula](/rules/geometric-series-formula), and practice [choosing a test](/learn/calculus-2/sequences-and-series/choosing-a-test).
- **Review 31A derivatives** before the quarter, since integration is differentiation in reverse.
- **Use the [formula sheet](/formulas)** and timed [practice exams](/practice), and check the [common calculus mistakes](/blog/common-calculus-mistakes) list before midterms.

## Frequently asked questions

### What is UCLA Math 31B?

Math 31B, Integration and Infinite Series, covers transcendental functions, methods and applications of integration, and sequences and series. In the current department outline it runs from Riemann sums and the Fundamental Theorem through integration techniques to convergence tests, power series and Taylor series.

### Do I need Math 31B before Math 32A?

No. Math 32A requires only Math 31A. Math 32B requires both Math 31B and Math 32A.

### Is Math 31B hard?

Many students find it harder than 31A, because integration requires choosing a technique and series require reasoning about infinite sums. Regular practice with mixed problems is the best preparation.

### Can I take both Math 31B and Math 3B?

No. Math 31B is not open for credit to students who have credit for Math 3B, the second life sciences calculus course.

### How should I prepare for Math 31B?

Make derivative rules and antiderivatives automatic, including exponential, log and inverse trig functions, since every integration technique relies on recognizing them.