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 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 . 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 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!