Simple But Evil #1 — Integrals To Test Your Calculus Students

This article is the first in a new series of “Simple But Evil” articles [and hopefully soon, videos]. The idea of the “Simple But Evil” series is to help your students [or you, if you are the student] fill in some inevitable knowledge gaps in math content.

This article will be about “Simple But Evil” integrals with the purpose of closing some Algebra and Pre-Calculus gaps. A typical Calculus I course covers integration techniques up to \(u\)-substitution. In Calculus II, a larger set of integration techniques are introduced. These are typically integration by parts, trig integrals, trig substitutions, partial fraction decomposition, and working with improper integrals. Within these broad topics, there are numerous tricks [not the bad kind aka gimmicks] that help to simplify matters. However, if you’ve taught a Calculus sequence you know that when it comes time to techniques of integration, there can be cognitive overload for students. Part of this overload is a result of knowledge gaps with content covered in Algebra and Pre-Calculus.

Below is a list of simple but evil integrals to help close these knowledge gaps. The purpose of this is to expose them to new patterns that often get overlooked in a Calculus course in a similar way that fractions are overlooked in Algebra. That is, these are problems that, while in textbooks in some form, are not emphasized enough. It’s as much a matter of giving these problems as it is in giving them in a good sequential exposition. Techniques of integration are often taught disjointed without revisiting previously taught techniques. The initial separation is fine, but eventually, students have to be able to mix concepts from previous courses or just even from the previous week.

This article is a “living” article. I will be updating it from time to time as I rediscover things from one class to another or from one student to another.

Now, without further ado …

Constantly Confusing

With high confidence, the majority of you will have the majority of your students get the majority of these problems wrong. Every one of these problems can be integrated directly. The stumbling block for many students is that they haven’t had enough exposure to \(e\), \(\pi\), and other versions of constants thrown together the way I’ve put them here. There is a chronic misreading and misunderstanding of \(e\) versus \(e^{x}\), for example.

  1. \(\displaystyle \int 3 \ dx \)
  2. \(\displaystyle \int e \ dx \)
  3. \(\displaystyle \int \pi \ dx \)
  4. \(\displaystyle \int e^{\pi} \ dx \)
  5. \(\displaystyle \int \pi e \ dx \)
  6. \(\displaystyle \int \frac{1}{e} \ dx \)
  7. \(\displaystyle \int \frac{\pi}{e} \ dx \)
  8. \(\displaystyle \int e^{2} \ dx \)
  9. \(\displaystyle \int e^{e} \ dx \)
  10. \(\displaystyle \int ex \ dx \)
  11. \(\displaystyle \int x^{e} \ dx \)
  12. \(\displaystyle \int x^{-\pi} \ dx \)
  13. \(\displaystyle \int x^{e + \pi} \ dx \)
  14. \(\displaystyle \int e^{\pi} x \ dx \)
  15. \(\displaystyle \int e e^{-x} \ dx \)
  16. \(\displaystyle \int \pi e^{x} \ dx \)
  17. \(\displaystyle \int \cos(1) \ dx \)
  18. \(\displaystyle \int \sin(\cos(1)) \ dx \)
  19. \(\displaystyle \int \sin(\cos(1))x \ dx \)
  20. \(\displaystyle \int e^{\sin(\cos(1))} \ dx \)
  21. \(\displaystyle \int z \ dx \)

You get the idea. Mix and match these constants and toss in a multiplication by \(x\) or \(x^{n}\) and you’ll see who sees the integrands correctly.

Similar But Different

Give these three integrals in the order given as one exercise once you’ve covered integration by parts. Too often, students see a polynomial with an exponential and jump immediately to integration by parts because integration by parts exercises are filled with these types. But as you can see, the third integral is a \(u\)-substitution. While an integral like the third one is covered when introducing \(u\)-substitution, the form is quickly forgotten. You can add another layer of difficulty by changing the exponent to \(ax\) or \(ax^{2}\) for \(a \neq 0\).

  1. \(\displaystyle \int xe^{x}\ dx\)
  2. \(\displaystyle \int x^{2}e^{x}\ dx\)
  3. \(\displaystyle \int xe^{x^{2}}\ dx\)

It’s Natural

Natural logs are confusing. In the first, the integrand is a constant, the second and third are integration by parts, the fourth and fifth are \(u\)-substitutions. For the second problem, students often can’t see that “\(dv = dx\)” is the choice in integration by parts. Also, pay close attention to your students in the fourth and fifth problems. Many may get the fourth problem correct, but then they’ll botch the fifth one because they got in their head that \(\frac{d[\ln(f(x))]}{dx} = \frac{1}{f(x)}\) — a common mistake.

  1. \(\displaystyle \int \ln(4) \ dx \)
  2. \(\displaystyle \int \ln(x) \ dx \)
  3. \(\displaystyle \int x\ln(x) \ dx \)
  4. \(\displaystyle \int \frac{\ln(x)}{x} \ dx \)
  5. \(\displaystyle \int \frac{2x\ln(x^{2}+1)}{x^{2}+1} \ dx \)

Not The Fractions!

Fractions still remain a hopeless mess for a number of students. Put them together with integrals and you have chaos that rivals government politics. The first problem tends to be more confusing than the second problem even though they are identical. Vary this by using \(e^{x}\) or \(\sin(x)\) or any other elementary function in place \(x\).

For sake of completeness, partial fractions are a disaster for students and there are a variety of simple but evil integrals. However, fractions by themselves are enough to befuddle students. So, the standard battery of partial fraction integrals are often enough evil!

Here is very simple but slightly evil involving fractions.

  1. \(\displaystyle \int \frac{x}{2} \ dx \)
  2. \(\displaystyle \int \frac{1}{2}x \ dx \)

The Arguments of Functions

Students get confused about the difference between \(f(x)\), \(f(2x)\), and \(2f(x)\). These are concepts that should’ve been clarified in a pre-Calculus course and resolved completely by the first part of a Calculus I course, but alas, there are cracks through which things slip. What brings these misunderstandings to bear is some trigonometry. Toss in some fractions for good measure.

  1. \(\displaystyle \int \sin(x) \ dx \)
  2. \(\displaystyle \int \frac{1}{2}\sin(x) \ dx \)
  3. \(\displaystyle \int \sin\Big(\frac{1}{2}x\Big) \ dx \)

It’s Too Radical

These integrals begin to confuse students once they are introduced to integration by trig substitution because of the hyper-conditioning on the \(\sqrt{x^{2} \pm a^{2}}\) or \(\sqrt{a^{2}-x^{2}}\) forms. There are other more elementary mistakes students make like \(\sqrt{x^{2} + a^{2}} = x + a\), but that’s a whole ‘nother problem.

  1. \(\displaystyle \int x^{2}+1 \ dx \) — I have seen enough students try \(x = \tan(\theta)\). It’s the long way around the barn.
  2. \(\displaystyle \int \frac{x}{\sqrt{x^{2}+1}} \ dx \) — They miss the \(u\) substitution in favor of \(x = \tan(\theta)\)
  3. \(\displaystyle \int \frac{x^{3}}{\sqrt{x^{2}+1}} \ dx \) — They miss the \(u\) substitution in favor of \(x = \tan(\theta)\)
  4. \(\displaystyle \int \frac{1}{\sqrt{-x^{2}+1}} \ dx \) — The negative sign in front of \(x^{2}\) effectively forbids them from seeing a standard inverse trig integral or the substitution \(x = \sin(\theta)\)

I’ll save improper integrals in a separate write-up. In the meanwhile, try these with your class and let me know how it goes!

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2 thoughts on “Simple But Evil #1 — Integrals To Test Your Calculus Students

  1. Joseph

    Bravo. I will be eagerly awaiting the rest of this series.

    In regards to the “constantly confusing” set: I frequently ask my calculus students to take the derivative of a weird constant, like e^pi. Somehow, it never occurred to me that I should have them integrate a weird constant. I’ll have to give that a try sometime.

    Reply

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