UIUC MATH 231 Calculus II: Topics, Study Guide and Practice

UIUC MATH 231 is Calculus II at the University of Illinois Urbana-Champaign. Students take it after either version of Calculus I, MATH 220 or MATH 221, and it leads into MATH 241. It is a 3-hour course built around four topics: integration techniques, conic sections, polar coordinates and infinite series. This guide maps each one to free practice with instant feedback.

What UIUC MATH 231 covers

According to the Illinois course catalog, MATH 231 is the second course in calculus and analytic geometry. It covers techniques of integration, conic sections, polar coordinates and infinite series, and it counts toward the Quantitative Reasoning I general education requirement.

That lines up closely with a standard Calculus 2 course. The integration half extends the u-substitution you learned in Calculus I to integration by parts, trig integrals, trig substitution and partial fractions. The series half moves from sequences to convergence tests and on to power and Taylor series. Conic sections and polar coordinates give the course its analytic geometry side.

MATH 231 topic map

Each row links to a lesson you can practice and to a reference page or guide that explains the idea.

Improper integrals are not named in the catalog description, but convergence of improper integrals is the natural bridge to the integral test for series, so it is worth practicing alongside the series material.

Where MATH 231 sits in the UIUC sequence

If you need to refresh Calculus I first, see the MATH 221 guide or the MATH 220 guide. For what comes next, see the MATH 241 guide.

Example problems at the MATH 231 level

1. Integration by parts. Evaluate \int x\cos x\,dx with u = x and dv = \cos x\,dx:

2. Partial fractions. Since \dfrac{1}{x^2 - 1} = \dfrac{1}{2}\left(\dfrac{1}{x - 1} - \dfrac{1}{x + 1}\right),

3. A geometric series.

The series converges because the ratio r = \frac{1}{3} satisfies |r| < 1.

How to do well in UIUC MATH 231

• Sort integrals before you solve them. Most mistakes come from picking the wrong technique. Work through choosing a strategy until you can name the method at a glance.
• Learn the series tests as a decision list. Start with the geometric series formula and the nth-term test, then add comparison, ratio and alternating tests.
• Memorize a handful of Maclaurin series. Knowing e^x, \sin x, \cos x and \frac{1}{1 - x} by heart saves time on every Taylor series problem. The formula sheet lists them.
• Practice mixed sets under time pressure. Exams mix topics, so finish each unit with practice exams that shuffle problems.

Tip: if a series has factorials or powers like 3^n, try the ratio test first. It settles most of those problems in a few lines.