# Penn State MATH 141 Calculus II: Topics, Study Guide, Practice

> A study guide for Penn State MATH 141 (Calculus with Analytic Geometry II): integration techniques, series, Taylor polynomials, parametric and polar.

Source: https://practicecalculus.com/courses/penn-state-math-141
Updated: 2026-11-03

## At a glance

- **Course code:** MATH 141 (honors version MATH 141H)
- **Credits:** 4
- **Prerequisites:** MATH 140 (or MATH 140A, 140B, 140E, 140G or 140H)
- **Covers:** Logarithms, exponentials and inverse trig functions, applications of the definite integral, integration techniques, sequences and series, power series and Taylor polynomials, parametric and polar
- **General education:** Quantification (GQ)
- **Next course:** MATH 230 (Calculus and Vector Analysis)
- **Practice here:** [Calculus 2](/learn/calculus-2)

## Key takeaways

- MATH 141 is Penn State's 4-credit Calculus II course, the second in a two- or three-course sequence for science and engineering.
- It opens with logarithms, exponentials and inverse trig functions, which Penn State places in Calculus II.
- The core is integration techniques and applications, then sequences, series, power series and Taylor polynomials.
- It ends with parametric equations and polar functions, and leads to MATH 230.

**Penn State MATH 141**, Calculus with Analytic Geometry II, is the second course in Penn State's calculus sequence for science, engineering and related fields. Worth 4 credits, it builds on MATH 140 with new functions, new integration techniques and infinite series. This guide maps every MATH 141 topic to practice so you can learn each one by solving problems with instant feedback.

## What Penn State MATH 141 covers

The bulletin lists the MATH 141 topics in order: logarithms, exponentials and inverse trigonometric functions; applications of the definite integral and techniques of integration; sequences and series; power series and Taylor polynomials; and parametric equations and polar functions.

Most of that is a standard **Calculus 2** course. The first unit is the exception: calculus of $e^x$, $\ln x$ and inverse trig functions, which many universities teach in Calculus 1. If you are coming to Penn State with credit from elsewhere, check that you have seen these.

## MATH 141 topic map

| MATH 141 topic | Practice | Read |
| --- | --- | --- |
| Logarithms and exponentials | [Exponentials and logarithms](/learn/calculus-1/derivative-rules/exponential-and-log-derivatives) | [Derivatives of exponential and log functions](/blog/derivatives-of-exponential-and-logarithmic-functions), [natural logarithm](/functions/natural-logarithm) |
| Inverse trig functions | [Chain rule: any function](/learn/calculus-1/derivative-rules/chain-rule-general) | [Arctangent](/functions/arctangent), [arcsine](/functions/arcsine), [inverse function rule](/rules/inverse-function-rule) |
| L'Hôpital's rule | [L'Hôpital's rule](/learn/calculus-1/applications-of-derivatives/lhopitals-rule) | [L'Hôpital's rule](/blog/lhopitals-rule) |
| Applications of the definite integral | [Arc length and surface area](/learn/calculus-2/improper-integrals-and-applications/arc-length-and-surface-area) | [Volumes of revolution](/blog/volumes-of-revolution) |
| Integration by parts | [Integration by parts](/learn/calculus-2/integration-techniques/integration-by-parts) | [Integration by parts](/blog/integration-by-parts) |
| Trig integrals and substitution | [Powers of sine and cosine](/learn/calculus-2/integration-techniques/trig-integrals) | [Trig substitution](/blog/trig-substitution) |
| Partial fractions | [Partial fractions](/learn/calculus-2/integration-techniques/partial-fractions) | [Partial fractions](/blog/partial-fractions) |
| Improper integrals | [Integrands that blow up](/learn/calculus-2/improper-integrals-and-applications/infinite-discontinuities) | [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](/blog/convergence-tests) |
| Power series | [Power series and radius of convergence](/learn/calculus-2/power-series-parametric-polar/power-series-and-radius) | [Power series](/blog/power-series) |
| Taylor polynomials | [Taylor polynomials](/learn/calculus-2/power-series-parametric-polar/taylor-polynomials) | [Taylor series](/blog/taylor-series), [Taylor's theorem](/theorems/taylors-theorem) |
| Parametric and polar | [Polar coordinates](/learn/calculus-2/power-series-parametric-polar/polar-coordinates) | [Parametric curves](/learn/calculus-2/power-series-parametric-polar/parametric-curves) |

## MATH 141 in the Penn State sequence

| Course | Credits | Role |
| --- | --- | --- |
| MATH 140 | 4 | Prerequisite: limits, derivatives, first integrals |
| MATH 141 | 4 | Transcendental functions, integration, series, polar |
| MATH 141H | 4 | Honors version of MATH 141 |
| MATH 230 | 4 | Next: multivariable and vector calculus |

Credit is allowed for only one of MATH 141, 141B and 141H. For a refresher on the first course, see the [MATH 140 guide](/courses/penn-state-math-140). For what comes next, see the [MATH 230 guide](/courses/penn-state-math-230).

## Example problems at the MATH 141 level

**1. An inverse trig derivative.** Since $\frac{d}{dx}\arctan x = \frac{1}{1 + x^2}$, the chain rule gives

$$\frac{d}{dx}\arctan(2x) = \frac{2}{1 + 4x^2}$$

**2. Partial fractions.** Evaluate $\int \frac{1}{x(x + 1)}\,dx$. Write $\frac{1}{x(x+1)} = \frac{1}{x} - \frac{1}{x + 1}$, so

$$\int \frac{1}{x(x + 1)}\,dx = \ln|x| - \ln|x + 1| + C$$

**3. A Taylor polynomial.** The degree 3 Maclaurin polynomial of $e^x$ is

$$P_3(x) = 1 + x + \frac{x^2}{2} + \frac{x^3}{6}$$

so $e^{0.1} \approx 1 + 0.1 + 0.005 + 0.000167 \approx 1.10517$.

## How to do well in Penn State MATH 141

- **Master the first unit fast.** Log, exponential and inverse trig derivatives show up inside every later integral. The [logarithmic differentiation rule](/rules/logarithmic-differentiation) is worth learning early.
- **Practice choosing a technique.** The [choosing a strategy lesson](/learn/calculus-2/integration-techniques/choosing-a-strategy) mixes problems the way exams do.
- **Learn a decision path for series.** Start with the [nth-term test](/rules/nth-term-test), then geometric and p-series, then the [ratio test](/rules/ratio-test).
- **Keep the [formula sheet](/formulas) close** for common Maclaurin series, and take timed [practice exams](/practice).
- **Review** [common calculus mistakes](/blog/common-calculus-mistakes) before each midterm.

> **Tip:** when a series has factorials or powers of $n$ in the exponent, try the ratio test first. It usually settles the question in two lines.

## Frequently asked questions

### What is MATH 141 at Penn State?

MATH 141, Calculus with Analytic Geometry II, is the second calculus course for science and engineering students. It covers logarithms, exponentials, inverse trig functions, applications of the definite integral, integration techniques, sequences and series, power series and Taylor polynomials, and parametric and polar functions.

### What do I need for MATH 141?

The enforced prerequisite is MATH 140 or one of its variants: MATH 140A, 140B, 140E, 140G or 140H.

### Is MATH 141 harder than MATH 140?

Many students find it harder, because integration requires choosing among several techniques and series introduce a new style of reasoning. Mixed practice, where you have to pick the method yourself, is the best preparation.

### What comes after MATH 141?

MATH 230, Calculus and Vector Analysis, which covers three-dimensional geometry, vectors, partial derivatives, multiple integrals and vector calculus. MATH 231 and MATH 232 split that material into two 2-credit courses.

### How do I study for MATH 141?

Start with exponential and log derivatives and substitution, since they feed every technique that follows. Then practice integration by parts and partial fractions until you can recognize which one a problem needs.