## A Surprising Integral

I had a fun encounter with an innocuous looking integral.

It all started with a simple directive: evaluate .

Integration is often tricky business. Although there is a large collection 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 . Thus, , and so . But since , we have .

This gives us .

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

After applying this technique, we’ll get . And so, after un-substituting, we get

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!

*Click here to see more in Appreciation.*

I like cos(sqrt(x)), because it looks like it should only be defined for positive x, but since cos(x) is even and analytic, you get a power series that works for all values of x. IIRC, the function defined by this series has a surprising graph for negative x.