Pattern Recognition

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 \cos^3x. The first step is recognizing that you could use substitution to find a solution. As \sin and \cos 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 \cos^3x can be written as \cos\cdot\cos\cdot\cos. And thus as \cos^2\cdot\cos as well. Let’s see:

    \[\int\cos^3 x\mathrm{d}x=\int\cos^2 x\cos x\mathrm{d}x\]

Now what? We also recognize the identity

    \[\sin^2+\cos^2\equiv 1\]

so

    \[\cos^2\equiv 1-\sin^2\]

Now let’s write it as

    \[\int(1-sin^2 x)\cos x\mathrm{d}x\]

And now it’s all a piece of cake:

    \[\sin x=u \Rightarrow du=\cos x\mathrm{d}x\]

and so

    \[\int (1-u^2)\mathrm{d}u=u-\tfrac{1}{3}u^3+C\]

Substitution gives us

    \[\sin x-\tfrac{1}{3}\sin^3 x+C\]

Cayce would be proud with rash.