# Berkeley Math 52 (formerly Math 1B) Calculus II: Study Guide

> A study guide for UC Berkeley Math 52, formerly Math 1B: integration techniques, sequences and series, and first and second-order differential equations.

Source: https://practicecalculus.com/courses/uc-berkeley-math-52
Updated: 2026-11-03

## At a glance

- **Course code:** Math 52 (formerly Math 1B)
- **Units:** 4 (three hours of lecture and three of discussion per week)
- **Offered:** Fall and spring
- **Prerequisites:** Math 51 (formerly 1A), N1A or equivalent
- **Covers:** Techniques and applications of integration, infinite sequences and series, first-order ordinary differential equations and second-order linear ordinary differential equations
- **Next course:** Math 53 (Multivariable Calculus) or Math 54 (Linear Algebra and Differential Equations)
- **Practice here:** [Calculus 2](/learn/calculus-2)

## Key takeaways

- Math 52 is the new number for Berkeley's Math 1B, the second semester of single-variable calculus.
- It covers integration techniques, a unit on infinite sequences and series, and first and second-order differential equations.
- The series unit gets the most class time in the official outline.
- The prerequisite is Math 51 (formerly 1A) or equivalent, and the course is offered in fall and spring.

**UC Berkeley Math 52**, long known as **Math 1B**, is the second semester of single-variable calculus at Berkeley. It picks up where [Math 51](/courses/uc-berkeley-math-51) (formerly 1A) ends and goes further than many Calculus 2 courses by including both first and second-order differential equations. This guide maps every Math 52 topic to free practice with instant feedback.

## What Berkeley Math 52 covers

The Berkeley math department describes Math 52 as a continuation of Math 51 covering techniques of integration and applications of integration, infinite sequences and series, first-order ordinary differential equations, and second-order linear ordinary differential equations. In the official outline, integration techniques take about nine class hours, a short unit covers further applications of integration, the sequences and series unit takes about twelve hours, and the two differential equations units take about eight hours each.

So Math 52 is a **Calculus 2** course without the parametric and polar unit, which Berkeley teaches at the start of Math 53, plus a solid introduction to differential equations.

## Math 52 (1B) topic map

| Math 52 topic | Practice | Read |
| --- | --- | --- |
| Integration by parts | [Integration by parts](/learn/calculus-2/integration-techniques/integration-by-parts) | [Integration by parts guide](/blog/integration-by-parts), [integration by parts rule](/rules/integration-by-parts-rule) |
| Trig integrals and trig substitution | [Trig substitution](/learn/calculus-2/integration-techniques/trig-substitution) | [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) | [U-substitution](/blog/u-substitution) |
| Further applications: arc length | [Arc length and surface area](/learn/calculus-2/improper-integrals-and-applications/arc-length-and-surface-area) | [Volumes of revolution](/blog/volumes-of-revolution) |
| Sequences | [Sequences and their limits](/learn/calculus-2/sequences-and-series/sequences) | [Limits at infinity](/blog/limits-at-infinity) |
| Series and convergence tests | [Comparison tests](/learn/calculus-2/sequences-and-series/comparison-tests) | [Convergence tests](/blog/convergence-tests), [ratio test](/rules/ratio-test) |
| Alternating series | [Alternating series](/learn/calculus-2/sequences-and-series/alternating-series) | [Alternating series test](/rules/alternating-series-test) |
| Power series and Taylor series | [Power series and radius](/learn/calculus-2/power-series-parametric-polar/power-series-and-radius) | [Power series](/blog/power-series), [Taylor series](/blog/taylor-series) |
| First-order differential equations | [Separable differential equations](/learn/calculus-2/improper-integrals-and-applications/separable-equations) | [Natural exponential function](/functions/natural-exponential-function) |

Second-order linear differential equations do not have a dedicated lesson yet. They rely on the exponential and trig derivatives from [Calculus 1](/learn/calculus-1/derivative-rules/exponential-and-log-derivatives).

## Where Math 52 fits at Berkeley

| Course | Former number | Main content |
| --- | --- | --- |
| Math 51 Calculus I | 1A | Limits, derivatives, basic integrals |
| Math 52 Calculus II | 1B | Integration techniques, series, differential equations |
| Math 53 Multivariable Calculus | 53 | Parametric and polar curves, vectors, partial derivatives, multiple integrals, vector calculus |
| Math 54 Linear Algebra and Differential Equations | 54 | Linear algebra with differential equations |

Continue with the [Math 53 guide](/courses/uc-berkeley-math-53) for multivariable calculus.

## Example problems at the Math 52 level

**1. Trig substitution.** With $x = \sin\theta$, $dx = \cos\theta\,d\theta$ and $\sqrt{1 - x^2} = \cos\theta$:

$$\int_0^1 \sqrt{1 - x^2}\,dx = \int_0^{\pi/2}\cos^2\theta\,d\theta = \frac{\pi}{4}$$

That is the area of a quarter of the unit circle, which checks out.

**2. An alternating series.** $\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}$ converges by the alternating series test because $\frac{1}{n}$ decreases to $0$, even though $\sum \frac{1}{n}$ diverges. So it converges conditionally.

**3. A second-order linear equation.** Solve $y'' - 3y' + 2y = 0$. The characteristic equation $r^2 - 3r + 2 = 0$ has roots $r = 1$ and $r = 2$, so

$$y = C_1 e^{x} + C_2 e^{2x}$$

## How to do well in Berkeley Math 52

- **Decide on a technique before you write.** Exams mix integrals, so practice recognizing substitution, parts, trig substitution and partial fractions on sight with the [techniques practice](/learn/calculus-2/integration-techniques/techniques-practice) set.
- **Give series the most time.** It has the largest block in the outline. Learn the [geometric series formula](/rules/geometric-series-formula) and [nth-term test](/rules/nth-term-test) first, then build up to the ratio and comparison tests.
- **Check differential equation solutions by substituting them back.** It takes seconds and catches most errors.
- **Keep the [formula sheet](/formulas) close**, read the [common calculus mistakes](/blog/common-calculus-mistakes), and test yourself with timed [practice exams](/practice).

## Frequently asked questions

### Is Berkeley Math 52 the same as Math 1B?

Yes. The Berkeley math department lists the course as Math 52, formerly 1B. It continues Math 51, which was formerly Math 1A.

### What does Math 52 (1B) cover?

It continues Math 51 with techniques and applications of integration, then covers infinite sequences and series, first-order ordinary differential equations, and second-order linear ordinary differential equations.

### Is Math 52 (1B) hard?

Series is the unit many students find hardest, since it asks whether infinite sums converge rather than for a single answer. The course also adds differential equations, which many Calculus 2 courses elsewhere leave out, so it covers a lot of ground.

### What comes after Math 52?

The next lower division courses are Math 53, Multivariable Calculus, and Math 54, Linear Algebra and Differential Equations. Both list the 1A and 1B sequence, now Math 51 and 52, or an equivalent as the prerequisite.