UT Austin M 408L Integral Calculus: Study Guide and Practice
A study guide for UT Austin M 408L, Integral Calculus: integration, the fundamental theorem, transcendental functions, sequences and series, with practice.
UT Austin Updated 3 min read
At a glance
- Course code
- M 408L (also offered as the 3-hour M 308L)
- Hours
- 4 for M 408L (three lecture and two discussion hours a week); 3 for M 308L
- Prerequisites
- M 408C, 408K or 408N with at least a C-, or M 408R or 408Q with at least a B
- Covers
- Integration, the fundamental theorem of calculus, transcendental functions, sequences and infinite series
- Credit overlap
- Only one of M 403L, 408L (or 308L) and 408S may be counted
- Next course
- M 408M (Multivariable Calculus) or M 408D
- Practice here
- Calculus 1 integrals units, then Calculus 2
On this page
M 408L is UT Austin's one-semester course in integral calculus. Students usually take it after M 408K, Differential Calculus, and it carries them from the first antiderivative through infinite series. This guide maps every M 408L topic to free practice, with instant feedback on each answer.
What UT Austin M 408L covers
The UT Austin catalog describes M 408L as an introduction to the theory and applications of integral calculus of one variable. Its listed topics are integration, the fundamental theorem of calculus, transcendental functions, sequences and infinite series.
Mapped onto a standard sequence, M 408L starts with the integral half of Calculus 1 (antiderivatives, Riemann sums, the fundamental theorem) and finishes with the core of Calculus 2 (integration techniques, sequences and series). The same course is offered as M 308L, which carries 3 hours instead of 4.
M 408L topic map
| M 408L topic | Practice | Read |
|---|---|---|
| Antiderivatives | Undoing derivatives | Antiderivatives, reverse power rule |
| Riemann sums and the definite integral | Area with rectangles | Riemann sums, definite integral properties |
| The fundamental theorem | Derivatives of integrals | FTC part 1, FTC part 2 |
| Substitution | U-substitution | u-substitution |
| Transcendental functions | Sine, cosine, eˣ and 1/x | Natural exponential function, inverse trig: arctangent |
| Area and volume | Area between curves | Area between curves, volumes of revolution |
| Integration by parts | Integration by parts | Integration by parts |
| Other techniques | Choosing a strategy | Partial fractions, trig substitution |
| Improper integrals | Integrating to infinity | Improper integrals |
| Sequences | Sequences and their limits | L'Hôpital's rule |
| Infinite series | Series and partial sums | Geometric series, nth-term test |
| Convergence tests | Ratio and root tests | Convergence tests, alternating series test |
Where M 408L fits at UT Austin
| Course | Title | Covers |
|---|---|---|
| M 408K | Differential Calculus | Limits, continuity, derivatives, mean value theorem |
| M 408L | Integral Calculus | Integration, transcendental functions, sequences, series |
| M 408M | Multivariable Calculus | Vectors, partial derivatives, gradients, multiple integrals, vector calculus |
| M 408C | Differential and Integral Calculus | M 408K topics plus integration, in one semester |
M 408L follows the M 408K guide and leads into the M 408M guide. Students from the accelerated M 408C course may also take M 408L, and M 408L in turn meets the prerequisite for M 408D.
Example problems at the M 408L level
1. The fundamental theorem, part 1. If F(x) = \int_0^x \sqrt{1 + t^3}\,dt, then
2. A substitution with a transcendental function. Evaluate \int \frac{2x}{x^2 + 1}\,dx with u = x^2 + 1, du = 2x\,dx:
3. A p-series. The series \sum_{n=1}^{\infty} \frac{1}{n^2} converges because it is a p-series with p = 2 > 1, while \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} diverges because p = \frac{1}{2} \le 1.
How to do well in UT Austin M 408L
- Start with a strong antiderivative base. Every technique later in the course reduces to a basic integral, so make the reverse power rule and the standard trig and exponential integrals automatic.
- Learn to choose a technique. The choosing a strategy lesson trains the decision, not just the steps.
- Give series extra time. Series reasoning is new. Work through the convergence tests guide and practice one test at a time before mixing them.
- Know the common errors. Our list of common calculus mistakes includes forgotten constants and dropped du terms.
- Use the formula sheet and timed sets. Keep the formula sheet close and finish each unit with practice problems.
Practice it now
Undoing derivatives
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