A Surprising Integral

I had a fun encounter with an innocuous looking integral.

It all started with a simple directive:  evaluate \int{cos(\sqrt{x}) \thinspace dx}.

Integration is often tricky business.  Although there is a large body of integration techniques, there isn’t really one guaranteed procedure for evaluating an integral.  If you see what the answer is, you write it down; if you don’t, you try a technique in the hope that it makes you see what the answer is.  If that technique doesn’t work, you try another.

This particular problem is interesting in that it highlights a strange phenomenon that occasionally pops up in problem-solving:  sometimes making a problem look more complicated actually makes it easier to solve.

Let u = \sqrt{x}.  Thus, du = \frac{1}{2\sqrt{x}} dx, and so dx = 2 \sqrt{x} du.  But since u = \sqrt{x}, we have dx = 2 \thinspace u \thinspace du.

This gives us \int{cos(\sqrt{x}) \thinspace dx} = \int{2 \thinspace u \thinspace cos(u) \thinspace du}.

This actually looks a bit more difficult than the original problem, but now we can easily integrate using Integration by Parts!

After applying this technique, we’ll get \int{2 \thinspace u \thinspace cos(u) \thinspace du} = 2 \thinspace u \thinspace sin(u) + 2 \thinspace cos(u) + C.  And so, after un-substituting, we get

\int{cos(\sqrt{x})} = 2\sqrt{x} \thinspace sin(\sqrt{x}) + 2 \thinspace cos(\sqrt{x}) + C.

I was surprised that this technique worked, so I actually differentiated to make sure I got the correct answer.  You can take my word for it, or you can verify with WolframAlpha.

One of the best parts of being a teacher is learning (or re-learning) something new every day!

TEDxNYED: Creativity and Mathematics

Here is the video of my talk at this year’s TEDxNYED conference.  My talk was on Creativity and Mathematics.

Mathematics is an inherently creative activity.  Students and teachers alike often fail to appreciate just how creative math really is, so I wanted to share some of the simple ways that students and I create with mathematics in our classroom.

Speaking at this year’s TEDxNYED conference was a professional highlight for me.  Getting to share ideas with so many interesting and passionate people was an honor, and the unique experience created by the speakers, attendees, and conference organizers was a true inspiration.

Inconsistent Temperatures

There are at least two things making me uncomfortable in this screenshot from “Caribbean One” TV.

The first disconcerting thing is mathematical in nature.  The top number in each column is the daily high temperature in degrees Celsius; the bottom number is the daily high temperature in degrees Fahrenheit.  Things don’t seem to add up.

The second disconcerting thing is the stuffed parrot, which I think needs no further explanation.

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