UCLA MATH 31B: Integration and Infinite Series

MATH 31B covers integration techniques, applications, improper integrals, and infinite sequences and series including Taylor series. It's the second course in UCLA's main calculus sequence and a prerequisite for the multivariable courses that follow.

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What makes it hard

Series in a quarter system is the notorious combination: the most abstract topic in lower-division calculus, compressed into the final weeks before a cumulative final. 31B is widely cited by UCLA students as the hardest course in the 31/32 sequence, and exam medians reflect it.

What you'll cover

• Integration by parts and partial fractions
• Trigonometric integrals and substitution
• Improper integrals
• Sequences and infinite series
• Convergence tests
• Taylor and Maclaurin series

The MATH 31B study guide

How to study for UCLA MATH 31B, step by step.

1
Make integration automatic in the first five weeks
Students who struggle in MATH 31B are usually still shaky on integrals when series arrives. Daily technique drills early — parts, partial fractions, trig substitution — free your full attention for the back half.
2
Give series everything when it starts
Series is the most abstract topic in lower-division calculus, compressed into the final weeks of a ten-week quarter before a cumulative final. Classify several series daily from the moment the unit opens.
3
Build a convergence-test selection routine
Exam questions reward picking the right test fast. Practice classifying many series quickly rather than perfecting a few — speed of correct selection is the graded skill.
4
Memorize the standard Taylor series cold
The expansions for e^x, sin, cos, and the geometric series are assumed instant on the final. Flashcard them early so exam time goes to the hard parts.
5
Front-load it all with Fennie
Upload the MATH 31B syllabus and Fennie's Daily Plans schedule integration practice early so you reach series with time to spare, with convergence-test quizzes and Taylor series flashcards generated from your actual materials. Free to start.
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How Fennie helps with MATH 31B

Fennie's Daily Plans front-load integration technique practice so you arrive at the series unit with time to spare — the students who struggle are the ones still shaky on integrals when convergence tests arrive. Chat through which convergence test applies and why, and drill flashcards on the standard Taylor series the final expects from memory.

FAQ

Is MATH 31B the hardest lower-division math class at UCLA?

It's the most common answer among students. Integration techniques demand pattern fluency, and the series unit is abstract material on a compressed timeline. The combination of both in ten weeks is what earns the reputation.

How do I pass MATH 31B?

Build integration fluency early through daily practice, so series gets your full attention in the back half. For convergence tests, practice classifying many series quickly rather than perfecting a few — exam questions reward fast correct test selection.

What comes after MATH 31B?

MATH 32A (multivariable differential calculus) for most STEM majors, with 32B and the 33 series after. 31B's series and integration skills return directly in 32B and differential equations.

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Upload your MATH 31B materials and Fennie generates a Daily Plan paced to your deadline — plus chat, flashcards, and quizzes built from the actual course content.

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More UCLA courses

MATH 31A — Differential and Integral Calculus

MATH 31A is UCLA's first calculus course — limits, derivatives, optimization, and the basics of integration — required across engineering, physical science, economics, and life-science-adjacent tracks. Most STEM freshmen take it (or skip it via AP credit) in fall quarter.

MATH 32A — Calculus of Several Variables

MATH 32A introduces multivariable calculus: vectors, vector-valued functions, partial derivatives, gradients, and optimization in several variables. It's required for engineering, physics, math, and CS-adjacent tracks, typically taken in the first year after the 31 sequence.

MATH 32B — Calculus of Several Variables

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MATH 33A — Linear Algebra and Applications

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