# Open-ended tasks for teaching students about integration techniques

One of the best algebra-teaching games I've seen is the "Four 4's" game, where students have to take 4 fours and construct every number from 1-100 using only those fours and algebraic operations:

44/44=1, 4/4+4/4=2, 4-(4/4)^4=3, etc.

This is a great way to get students to explore the power of algebra. It is open ended, requires creativity, and gets students to use a variety of algebraic operations.

Q:Is there anything similar for integration techniques?

By similar, I mean an easily-stated, open-ended task that requires students to be creative and gets them to use a variety of integration techniques.

I've thought of integration bee-type contests, but those aren't open ended.

• I am fond of some of the examples given here in Norton Starr's brief MAA note Estimating Definite Integrals in the College Mathematics Journal. I think estimation is somewhat overlooked in standard Calculus curricula, and that problems of this nature might allow for more of the "creative" approaches you are looking for. (Incidentally, I've always liked iv in the link!) – Benjamin Dickman Mar 20 '15 at 21:46

## 1 Answer

One activity is to ask them to do one problem in multiple ways. For example, "Find $\int \sin(x) \cos(x) dx$ in five ways". (Some possible ways are to use a trig identity, use integration by parts integrating cos first, use integration by parts integrating sin first, use substitution with $u=\sin(x)$, use substitution with $u=\cos(x)$.)

A favourite of mine is to ask to do $\int x^2 dx$ in multiple ways. My colleague calls this "stupid integration" (because you're not doing the most effective method).

One interesting upshot of this activity is that the results often look different, but you can show that they are actually the same, or at least only different by a constant. This is an important lesson that they often don't learn in ordinary "closed" integration questions.

• Doing the same thing multiple ways is an excellent strategy. – Karl Mar 21 '15 at 8:43
• "[O]r at least only different by a constant"—this deserves to be emphasised, I think. I love asking students precisely this question, to force them to realise that the answer to the problem is not just $\frac1 2\sin(x)^2$—that is, that the constant of integration really is part of the answer, not just a 'gotcha' way for the teacher to take off a point. – LSpice Mar 29 '15 at 19:59