The William Gibson novel Pattern Recognition tells the story of Cayce Pollard, allergic to logos and brands. Her allergy for these patterns make her a top consultant for many marketing companies. She would have done well in math, as she would have had rash all the time. Pattern recognition is valuable in mathematics. Not recognizing a mathematical pattern might mean a long and tedious way to the solution of your problem. Recognizing it might lead you to an instantaneous solution.
We want to find the antiderivative of . The first step is recognizing that you could use substitution to find a solution. As and are tightly locked together, you immediately start searching for a kind of substitution where one is the derivative or antiderivative of the other.
But it’s also nice to know that can be written as . And thus as as well. Let’s see:
Now what? We also recognize the identity
Now let’s write it as
And now it’s all a piece of cake:
Substitution gives us
Cayce would be proud with rash.