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 
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 
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!
4 Comments
Joshua Bowman (@Thalesdisciple) · June 8, 2012 at 10:34 pm
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.
Timothy Gowers · February 27, 2013 at 8:59 am
A technique I like that works here is “successive approximation”. Suppose we start with a naive guess of sin(sqrt(x)), chosen because the sin will at least differentiate to the cos. Then the chain rule gives us an unwanted factor of (2x)^{-1/2}. So to compensate for that we go for (2x)^{1/2}sin(sqrt(x)). But now the product rule gives us an unwanted term of x^{-1/2}sin(sqrt(x)). At this point we can observe that the factor x^{-1/2} is exactly what we need to make the unwanted term easy to integrate.
MrHonner · February 28, 2013 at 11:33 am
Wow. That’s really cool. It strikes me like integration by parts from another perspective: you are essentially replacing one integral with the sum of a function and another integral, hoping that the new integral is easier to compute, and you find it by playing around with the product rule and chain rule.
I just tried to use this technique to integrate f'(g(x)) in general, and noticed that the “second” approximation will work when g”(x) / (g'(x))^3 is a constant. This is precisely the situation we have here with g(x) = sqrt(x).
Drew Armstrong · January 13, 2017 at 8:53 pm
I agree that the best part of teaching is learning.