# Princeton MAT 104 Calculus II: Syllabus, Topics and Practice

> A study guide for Princeton MAT 104 Calculus II: integration techniques, polar coordinates, improper integrals, series and Taylor's theorem, with practice.

Source: https://practicecalculus.com/courses/princeton-mat-104
Updated: 2026-11-08

## At a glance

- **Course code:** MAT 104
- **Credits:** Not stated on the department course page
- **Offered:** Not stated on the department course page
- **Prerequisites:** MAT 103 or equivalent, plus calculator-free fluency with the standard library of functions
- **Covers:** Techniques and applications of integration, an introduction to differential equations, polar coordinates, improper integrals, infinite series and Taylor's theorem with remainder
- **Next course:** MAT 201 (Multivariable Calculus)
- **Practice here:** [Calculus 2](/learn/calculus-2)

## Key takeaways

- MAT 104 is the second semester of Princeton's standard three-semester calculus sequence.
- It covers integration techniques and applications, an introduction to differential equations, polar coordinates, improper integrals, series and Taylor's theorem.
- The department says no AP or IB score is equivalent to passing MAT 104, not even a 5 on AP Calculus BC.
- Problems often combine several techniques, so fluency with algebra and the standard functions is essential.

**Princeton MAT 104** is Calculus II, the second semester of Princeton's standard three-semester calculus sequence. It builds on [MAT 103](/courses/princeton-mat-103) with integration techniques, series and Taylor approximation, topics the department calls fundamental to the natural sciences, engineering and finance. This guide maps each MAT 104 topic to free practice with instant feedback.

## What MAT 104 covers

The math department's course page lists these topics: techniques and applications of integration, including area, length and volume; an introduction to differential equations; polar coordinates; analysis of convergence of infinite series and improper integrals; and approximation using Taylor's theorem with remainder.

The page also explains the style of the course. It emphasizes mature problem-solving skills through problems that need sophisticated pattern recognition and the ability to combine several techniques in one problem. As in MAT 103, no computational technology or internet resources are allowed on homework or exams.

## MAT 104 topic map

| MAT 104 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) |
| 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) |
| Choosing a technique | [Choosing a strategy](/learn/calculus-2/integration-techniques/choosing-a-strategy) | [U-substitution](/blog/u-substitution) |
| Area and volume | [Volumes with washers](/learn/calculus-1/applications-of-integrals/volumes-washers) | [Volumes of revolution](/blog/volumes-of-revolution) |
| Arc length | [Arc length and surface area](/learn/calculus-2/improper-integrals-and-applications/arc-length-and-surface-area) | [Volumes of revolution practice](/practice-problems/volumes-of-revolution) |
| Introduction to differential equations | [Separable equations](/learn/calculus-2/improper-integrals-and-applications/separable-equations) | [Differential equations practice](/practice-problems/differential-equations) |
| Polar coordinates | [Polar coordinates](/learn/calculus-2/power-series-parametric-polar/polar-coordinates) | [Parametric and polar practice](/practice-problems/parametric-and-polar) |
| Improper integrals | [Integrating to infinity](/learn/calculus-2/improper-integrals-and-applications/infinite-limits-of-integration) | [Improper integrals](/blog/improper-integrals) |
| Convergence of infinite series | [Comparison tests](/learn/calculus-2/sequences-and-series/comparison-tests) | [Convergence tests](/blog/convergence-tests) |
| Taylor's theorem with remainder | [Estimating with Taylor](/learn/calculus-2/power-series-parametric-polar/estimating-with-taylor) | [Taylor's theorem](/theorems/taylors-theorem), [Taylor series](/blog/taylor-series) |

## Placement: MAT 104 or something else?

The department compares MAT 104 to AP Calculus BC but is clear that test scores are not a reliable measure: a 5 on BC, a 7 on IB Math HL or an A at A-level is not equivalent to passing MAT 104. It suggests that two years of high school calculus with good grades may be comparable, and it uses a placement quiz early in the semester to help students confirm their choice.

Most students in MAT 104 are first-year students heading toward a natural science, engineering, math-track economics or finance. The prerequisite note is worth reading twice: the course expects you to solve equations, differentiate and sketch graphs with the standard functions quickly and without a calculator.

## Example problems at the MAT 104 level

**1. Integration by parts.**

$$\int x \cos x\,dx = x \sin x - \int \sin x\,dx = x \sin x + \cos x + C$$

**2. A convergence question.** The series $\sum_{n=1}^{\infty} \frac{1}{n^2 + 1}$ converges by comparison with the [p-series](/learn/calculus-2/sequences-and-series/integral-test-and-p-series) $\sum \frac{1}{n^2}$, since $\frac{1}{n^2+1} < \frac{1}{n^2}$.

**3. Taylor approximation.** Near $0$, $e^x \approx 1 + x + \frac{x^2}{2}$, so $e^{0.1} \approx 1.105$. Taylor's theorem bounds the error by roughly $\frac{e^{0.1}(0.1)^3}{6} < 0.0002$.

## How to do well in MAT 104

- **Keep a decision checklist for integrals.** Most exam integrals are hard because you must pick the method. Practice [choosing a strategy](/learn/calculus-2/integration-techniques/choosing-a-strategy) until it feels routine.
- **Learn the convergence tests as a set.** Work through [choosing a test](/learn/calculus-2/sequences-and-series/choosing-a-test) so you know which test fits which series.
- **Refresh MAT 103 skills early.** A quick pass through [u-substitution](/learn/calculus-1/integrals/u-substitution) and [inverse trig derivatives](/learn/calculus-1/derivative-rules/inverse-trig-derivatives) saves time later.
- **Practice under exam conditions** with the [practice exams](/practice), since calculators are not allowed.

After MAT 104, see the [MAT 201 guide](/courses/princeton-mat-201) for multivariable calculus.

## Frequently asked questions

### What is Princeton MAT 104?

MAT 104 is Calculus II, the second semester of Princeton's standard calculus sequence. It covers techniques and applications of integration, an introduction to differential equations, polar coordinates, improper integrals, infinite series and Taylor approximation.

### What are the prerequisites for MAT 104?

MAT 103 or equivalent. The department adds that you need expert, calculator-free familiarity with polynomials, rational and root functions, logarithms, exponentials, and trig functions and their inverses.

### Does a 5 on AP Calculus BC count as MAT 104?

No. The department says a 5 on BC, a 7 on IB Math HL or an A at A-level is not equivalent to a passing knowledge of MAT 104.

### Is MAT 104 hard?

The department says it emphasizes mature problem-solving through problems that need pattern recognition and several techniques at once. Many students find series and choosing an integration method the hardest parts.

### What comes after MAT 104?

MAT 201, Multivariable Calculus, is the third semester of the standard sequence. Some math majors go from MAT 104 to MAT 215 instead.