Calculus II and what to watch out for.

• Wm_Davies
In summary, Calc II is often considered the hardest course in the calculus sequence due to the wide range of topics it covers. Many people struggle with the large number of formulas and specific techniques that need to be learned for integration, series, and differential equations. However, some find it easier than Calc I because it is more focused on applying these techniques rather than understanding the basic concepts. While some of the material may not be used frequently in other math courses, it is still important to have a strong foundation in these concepts. Overall, the difficulty of Calc II may vary depending on the individual's learning style and major, but it is generally seen as a challenging course.
Wm_Davies
I hear that calc II is the hardest course in the calculus sequence. I am not sure why this is, but it is something that I hear constantly. Why is it considered to be so difficult? Are there a lot of formulas or are there many different applications of formulas, ideas, and logic? Obviously people will have different experiences with calc II, but I am really interested in hearing any opinion as to why this course is hard, easy, or of an average difficulty.

I thought it was actually quite a bit easier than Calc I, where you have come to grasp with the basic ideas,etc. Atleast where I took it, it is almost entirely about learning ways to solve integral and a few various applications: Diff Eqs. Polar Coords. Series. Thought it was a good class but not particularly difficult.

I had a hard time with solving triganomic integrals, work problems, and solving systems of DEs.

Topher925 said:
I had a hard time with solving triganomic integrals, work problems, and solving systems of DEs.

I'm finishing that course now, and trigonometric integrals have been tough for me. Overall, the course isn't exceptionally difficult. What has been the hardest for me is learning a large number of specific formulations and figuring out when each one should be used.

I thought it was the hardest in the Calculus sequence, by far.
I actually took it at the same time as Calc III (I was kind of learning Calc I at the same time as well) and Calc III was much easier and more intuitive...even though I was taking both at the same time. (So I was doing double and triple integrals at the same time as learning integration in Calc II.)

The other thing I've noticed is that I use the material from Calc II the LEAST out of any of the calculus sequence.
First of all, you rarely come across exotic integrals like the ones you learn in Calc II.
Second of all, if I do come across a difficult integral, I end up using other methods...like differentiating under the integral sign, using residue integration, etc.

Same thing with the methods we used for Series in Calc II...If I need to find out if a series converges, I pretty much only use the ratio test. Or, if I can't do that...there is some way to tweak the series by differentiating or integrating term by term, algebraically tweaking it into a better form, etc. I hope I don't get bit somewhere down the line, but I really don't remember much from Calc II. lol
I've come across the need for partial fractions in Complex variables, but then you learn that if you have a simple pole, you can differentiate the denominator...and if you have a higher order pole, you differentiate the whole function, etc.

The more I think about it...I don't know how much trouble I'd have if I never even took the course.
(I'm sure there's important things I'm not realizing right now, so I'm really only half serious)

Troponin said:
I thought it was the hardest in the Calculus sequence, by far.
I actually took it at the same time as Calc III (I was kind of learning Calc I at the same time as well) and Calc III was much easier and more intuitive...even though I was taking both at the same time. (So I was doing double and triple integrals at the same time as learning integration in Calc II.)

The other thing I've noticed is that I use the material from Calc II the LEAST out of any of the calculus sequence.
First of all, you rarely come across exotic integrals like the ones you learn in Calc II.
Second of all, if I do come across a difficult integral, I end up using other methods...like differentiating under the integral sign, using residue integration, etc.

Same thing with the methods we used for Series in Calc II...If I need to find out if a series converges, I pretty much only use the ratio test. Or, if I can't do that...there is some way to tweak the series by differentiating or integrating term by term, algebraically tweaking it into a better form, etc.

I hope I don't get bit somewhere down the line, but I really don't remember much from Calc II. lol
I've come across the need for partial fractions in Complex variables, but then you learn that if you have a simple pole, you can differentiate the denominator...and if you have a higher order pole, you differentiate the whole function, etc.

The more I think about it...I don't know how much trouble I'd have if I never even took the course.
(I'm sure there's important things I'm not realizing right now, so I'm really only half serious)

Here I think are the important concepts from Calc II that linger long after the class is over:

Solving simply ODEs
Integration by Parts
Polar Coordinates and Parametric curves
Some of the stuff to do with series

Of course your mileage my vary depending on your major etc. I found so far that things like integration by trig substitution and partial fractions, thankfully has yet to be seen again!

It seems that the general response is that integration by trig substitution is one of the harder methods to learn and master.

Wm_Davies said:
It seems that the general response is that integration by trig substitution is one of the harder methods to learn and master.

I actually think the computational stuff just gets so tedious and that really is the issue. The concepts are actually pretty easy to understand. My vote for most tedious method is definitely partial fractions though.

I learned that complex numbers/analysis.. make partial fractions super easy

Depending on your school, Calculus II can cover quite a breadth of material. There is the first phase where you cover all the basic methods of integration for various functions, and then you crash head on through sequences/series, methods for testing convergence, introductory ODEs, and even some taylor/fourier series material. We were expected to become fluid with our basic integration techniques, so we started with about a hundred integrals a week. Then the focus was on problem solving - which took a bit to wrap my head around how to approach certain problems using integration. We even covered some 3D material using single variable integration (utilizing symmetry).

As an engineering major, that semester was killer for me not just due to Calculus II but due to the combination of courses I took. Tacking on General Physics II, Linear Algebra, and General Chemistry II made for the most difficult semester I've experienced so far (I'm now a Junior). It's compounded by the fact that having a month off for winter break is nowhere near the same as three months off for summer - I always start catching fire earlier in the spring semester.

Just don't fall behind - the material moves quickly.

I didn't find integration particularly difficult, only much more tedious at times. Almost everything I learned in calculus two was used extensively in differential equations in one form or another. Many of the methods involve integrating all sorts of functions and it would be wise to have experience with infinite series for differential equations as well. I rarely need partial fractions for integrating; however, the same technique comes up when using Laplace transforms, again in differential equations. By and large calculus two was a very important prerequisite for my differential equations class.

As many said before, it kind of varies from place to place what "Calc 2" covers. In my experience, calc 2 was the most annoying and unsatisfying. About 2/3rds of the semester was spent solving extremely nasty trigonometric and exponential integrals using methods like substitution, integration by parts, etc. The last third of the class being about taylor and maclaurin series. Having graduated now (BS in Chemistry, math minor), I feel like calc 2 was one of the most useless. Even in my quantum chem class, the integrals we had to deal with were no where near as tedious and frustrating to solve than the ones we were exposed to in calc 2. Oh by the way, the general outline of the calc classes I took went something like this:
Calc 1: Limits, derivatives (general description in 2 dimensions, chain rule, product rule, etc), definition and simple examples of integrals
Calc 2: Esoteric integral HELL.
Calc 3: Basically Calc 1 in 3 dimensions - polar coordinates, partial derivatives (only difference to derivatives of a single variable function is that you're finding a tangent plane instead of a tangent line), some double and triple integrals (not nearly as terrible as it may sound)

Also I think most Calc IIIs contain a bit of vector calc: eg. line and surface integrals and various theorems involving them

Calc 2 is indeed the hardest course out of the calculus sequence. Personally, what made this course difficult was sequence and series (evaluating, integrating functions, finding limits of a function using series) then polar coordinates. Techniques of integration was not that bad, but every tool you learned to use to integrate a function comes back to you, you would be given an integral and asked to evaluate it, you would have to think how you would go around that, and most of the time a simple u-substitution, or a plug and chug formula would not do the job.

My prof for calc 2 was a guy who thought that problem solving was the most interesting thing in the world so he would concoct the strangest integrals he could to force us to use the material in subtle ways and he would often throw an esoteric integral out there once and you wouldn't see it again until the test (I couldn't find some of them in the book, which was somewhat problematic).

The series stuff was fun and pretty easy. Taylor polynomials are useful as hell and have some very interesting uses later in numerical analysis and in the power series ring (the series give a completion of the ring of polynomials over a field).

I don't think Cal II even close to ODE. As for Cal III, a lot of school don't cover Green's, Stoke's, Divergence, line integrals, then yes it is easier. But the theorems I mentioned above is another world. I know SJSU and the junior college in my area don't even touch it or just barely scratch the surface. BUt if you watch the MIT lecture on Youtube, they cover every bit of it. That make CAL II a child's play.

I think ODE is consider as CAL IV. That is a giant leap from the rest. The concept is totally different.

NO, I think the difficulty of the 4 calculus is exponential! I study most of Cal II and Cal III by myself, the only topics in Cal II that gave me a little difficulty is the series, other than that, the rest are not bad. AND you better study the power series well, you are going to need it in ODE and PDE. I currently studying PDE on my own, the Bessel, Lagendre and Strum Liouville are mostly using series.

As for Cal III. If people want to take EM class in the future, it is advisable to find a way to study line integral, Stoke's, Divergence and Green's theorem good. YOu are going to need it. They do go over these in the EM class. But believe me, you need all the help you can get in EM class, it would at least make it a little easier for yourself. You think these Cal classes are bad, wait until you study the EM class!

Last edited:
Wm_Davies said:
It seems that the general response is that integration by trig substitution is one of the harder methods to learn and master.

Not really, you see $$1-x^{2}$$ then think $$sin^{2}\theta + cos^{2}\theta = 1$$

You see $$1+x^{2}$$ think $$1+tan^{2}\theta = sec^{2}\theta$$!

1. What is Calculus II and why is it important?

Calculus II is the second course in the study of calculus, a branch of mathematics that focuses on rates of change and accumulation. It is important because it builds upon the concepts and techniques learned in Calculus I and is necessary for understanding more advanced mathematical topics and applications in fields such as physics, engineering, economics, and more.

2. What topics are covered in Calculus II?

Calculus II typically covers integration techniques, applications of integration, infinite series, and differential equations. Some courses may also include topics such as parametric and polar equations, vector calculus, and multivariable calculus.

3. What are some common challenges in Calculus II?

Many students struggle with the new concepts and techniques introduced in Calculus II, such as integration by parts, partial fractions, and infinite series. It can also be challenging to apply these concepts to real-world problems and to visualize and interpret the results.

4. How can I succeed in Calculus II?

To succeed in Calculus II, it is important to attend all lectures and actively participate in class. Practice regularly by solving problems and seeking help when needed. It is also helpful to create a study schedule and stay organized. Building a strong foundation in Calculus I can also make Calculus II more manageable.

5. What are some common mistakes to avoid in Calculus II?

One common mistake in Calculus II is not fully understanding the fundamental concepts and techniques, which can lead to errors in more complex problems. It is also important to carefully read and understand the problem before attempting to solve it. Additionally, it is important to check answers and units for accuracy.

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