# Harvard Math 1b Integration, Series and ODEs: Study Guide

> A study guide for Harvard Math 1b: integration techniques and applications, Taylor series and differential equations, with every topic mapped to free practice.

Source: https://practicecalculus.com/courses/harvard-math-1b
Updated: 2026-11-03

## At a glance

- **Course code:** Mathematics 1B (Math 1b)
- **Credits:** 4
- **Recommended prep:** Math 1a or an equivalent background
- **Covers:** Integration techniques and their applications, infinite series and Taylor series, and an introduction to ordinary differential equations, with a running theme of successive approximation
- **Next course:** Mathematics 21A (Multivariable Calculus) or Mathematics 19A for the life sciences
- **Practice here:** [Calculus 2](/learn/calculus-2)

## Key takeaways

- Math 1b is the second half of single-variable calculus at Harvard.
- It has three parts: integration, infinite series including Taylor series, and an introduction to differential equations.
- A central theme is successive approximation: finding approximate answers and controlling their error.
- Courses that list Math 1b as a prerequisite may expect mastery of all three parts.

**Harvard Math 1b** is the second semester of single-variable calculus at Harvard. Students arrive from [Math 1a](/courses/harvard-math-1a), from Math Ma and Mb, or straight from high school with a Math 1b placement. The course gives scientists and social scientists the working tools of calculus: integration, series and differential equations. This guide links every Math 1b topic to free practice with instant feedback.

## What Harvard Math 1b covers

The Harvard Mathematics Department lists three main areas for Math 1b: integration, infinite series, and differential equations. Integration is taken further than in 1a, with more advanced techniques and applications. Series lead to Taylor series, which represent functions as infinite polynomials. Differential equations are introduced as models of change in the physical, life and social sciences. Running through all three is the idea of successive approximation: when an exact answer is hard or impossible, find an approximate one, improve it, and keep track of the error.

In standard terms, Math 1b is **Calculus 2** with an introductory unit on differential equations.

## Math 1b topic map

| Math 1b topic | Practice | Read |
| --- | --- | --- |
| Substitution review | [U-substitution refresher](/learn/calculus-2/integration-techniques/u-substitution-refresher) | [U-substitution](/blog/u-substitution) |
| Integration by parts | [Integration by parts](/learn/calculus-2/integration-techniques/integration-by-parts) | [Integration by parts guide](/blog/integration-by-parts) |
| Trig substitution and partial fractions | [Partial fractions](/learn/calculus-2/integration-techniques/partial-fractions) | [Trig substitution](/blog/trig-substitution), [partial fractions](/blog/partial-fractions) |
| Applications: volume, work, mass | [Work](/learn/calculus-2/improper-integrals-and-applications/work) | [Volumes of revolution](/blog/volumes-of-revolution) |
| Improper integrals | [Integrating to infinity](/learn/calculus-2/improper-integrals-and-applications/infinite-limits-of-integration) | [Improper integrals](/blog/improper-integrals) |
| Sequences and series | [Series and partial sums](/learn/calculus-2/sequences-and-series/series-and-partial-sums) | [Geometric series](/blog/geometric-series) |
| Convergence tests | [Choosing a test](/learn/calculus-2/sequences-and-series/choosing-a-test) | [Convergence tests](/blog/convergence-tests) |
| Taylor series | [Common Maclaurin series](/learn/calculus-2/power-series-parametric-polar/maclaurin-series) | [Taylor series](/blog/taylor-series), [Taylor's theorem](/theorems/taylors-theorem) |
| Approximation and error | [Estimating with Taylor polynomials](/learn/calculus-2/power-series-parametric-polar/estimating-with-taylor) | [Linear approximation](/blog/linear-approximation) |
| Differential equations | [Separable differential equations](/learn/calculus-2/improper-integrals-and-applications/separable-equations) | [Exponential functions](/functions/exponential-functions) |

## Where Math 1b sits in the Harvard sequence

| Course | Level | Main content |
| --- | --- | --- |
| Math 1a | Calculus 1 | Derivatives and the basics of integration |
| Math 1b | Calculus 2 | Integration, series, differential equations |
| Math 21a | Calculus 3 | Multivariable calculus in two and three dimensions |
| Math 19a | Calculus 3 for life sciences | Multivariable calculus and differential equations for modeling |

The department recommends Math 1b as essential preparation for [Math 21a](/courses/harvard-math-21a). Students in biology-related concentrations should also read the [Math 19a guide](/courses/harvard-math-19a).

## Example problems at the Math 1b level

**1. Integration by parts.** With $u = \ln x$ and $dv = dx$:

$$\int \ln x\,dx = x\ln x - \int x \cdot \frac{1}{x}\,dx = x\ln x - x + C$$

**2. A Taylor approximation.** The degree 2 Maclaurin polynomial of $e^x$ is $1 + x + \frac{x^2}{2}$, so

$$e^{0.1} \approx 1 + 0.1 + 0.005 = 1.105$$

The true value is about $1.10517$, and the error is close to the next term, $\frac{0.1^3}{6} \approx 0.00017$.

**3. A separable differential equation.** Solve $\frac{dy}{dt} = 3y$ with $y(0) = 2$. Separate and integrate: $\int \frac{dy}{y} = \int 3\,dt$, so $\ln|y| = 3t + C$ and

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

## How to do well in Harvard Math 1b

- **Practice choosing a method.** Exams mix techniques, so drill the [choosing a strategy](/learn/calculus-2/integration-techniques/choosing-a-strategy) lesson until the right technique is obvious.
- **Build series intuition early.** Start with the [geometric series formula](/rules/geometric-series-formula) and the [ratio test](/rules/ratio-test), then work up to Taylor series.
- **Always estimate the error.** Successive approximation is a theme of the course, so get used to asking how close an approximation is.
- **Keep the [formula sheet](/formulas) handy**, review [common calculus mistakes](/blog/common-calculus-mistakes), and test yourself with timed [practice exams](/practice) before each midterm.

## Frequently asked questions

### What is Harvard Math 1b?

Math 1b, Integration, Series and Differential Equations, continues single-variable calculus after Math 1a. It covers more advanced uses of integration, infinite and Taylor series, and an introduction to ordinary differential equations, with a focus on modeling change in the sciences.

### What comes after Math 1b?

Most students continue to multivariable calculus in Math 21a. Students heading into a life science concentration often take Math 19a instead, which covers multivariable calculus and differential equations with biological applications.

### Is Math 1b hard?

Series is the part most students find new and abstract, because it asks whether infinite processes settle down rather than for a single number. Working many short convergence problems builds the intuition faster than rereading notes.

### Can I skip Math 1a and start in Math 1b?

Yes, if your Harvard Math Placement recommendation says Math 1b. Students who complete Math Ma and Mb also continue directly to Math 1b.

### How should I prepare for Math 1b?

Make derivatives, basic antiderivatives and u-substitution automatic, since every new technique in 1b builds on them. Then preview integration by parts and geometric series.