Integration is one of, if not the most, sophisticated of mathematical pleasures. It’s the ultimate adventure of problem-solving, engineering and creativity. It’s an elaborate dance of mathematical warfare. It’s a lifelong journey of learning and experience, upon which we are forever taking finite steps. It is, quite simply, an art.
There are few other places in maths where you will find such a tremendous overlap of mathematical wizardry – from completing the square, polynomial division and partial fractions, to exponentials, hyperbolic trigonometry, and complex numbers. This is, of course, a natural result of the fundamental nature of calculus – given a function, there are 3 questions we always tend to ask: What does it look like? What is its derivative? What is its integral?
I used to despise integration – oh, I thought I did – but no, what I despised was the thought of integration. If you only pass that initial hurdle of fear and unfamiliarity, you will find yourself atop a summit overlooking swathes of (continuous) landscapes awaiting exploration. The abstract rays of infinitesimal thickness will never have felt so vitalising upon your skin.
Integration is the pinnacle of mathematical problem solving. If you are a fan of thinking, then you will surely enjoy integration. If you like logic puzzles, then integration isn’t far off at all – you’ve just got a couple more symbols to worry about. If you’re into hyper-realistic real-time ray-traced 1st-person shooter games, then… eh, still a nonzero probability you’ll like integration.
If differentiation is but a water molecule, then integration is the ocean itself, with tides that shape the shore of your understanding, limitless depths of undiscovered beauty, and more chaos to float your boat than you could ever imagine. The uncertainty of the blue waters staring back at you may cast terror into your heart, but with nought but a few days at sea, it will soon start to feel like home.
As my friend iTechnical↗ famously alluded to, it’s rather hilarious how integrals are a kind of mathematical problem where almost all of the questions one could possibly construct are unsolvable. Of course, among the uncountably infinite number of potential problems, this still leaves more than enough to satiate our mathematical thirst. You take a tiny subset of the possible questions, and you still can’t run out. That enormity of the expanse that is integration is quite humbling to appreciate. It is not dissimilar from realising one’s insignificance within the universe.
The richness and diversity of integration is utterly unparalleled. Differentiation may be greyscale, but integration is OKLCH with no upper bound on chroma. Quadratics, cubics, quartics? Integrals are the whole frickin Taylor series.
Integration trains you in so much more than simply memorising antiderivatives or formulae. It teaches you to analyse, strategise, apply and simplify, surgically manipulating with every mathematical tool at your disposal. With enough time, integration becomes second nature.
Integration is to mathematics what organic synthesis is to chemistry: it opens up unimaginable new worlds of discovery and possibility. Integrals are the chords of calculus, the harmonies of its bewitching symphony. Integration
Y’know what, if you’re into real-time 1st-person shooter games, you’re gonna f***ing love integration.
Why integrate? Because it’s fun. It’s hard. It’s dynamic, unexpected and mysterious. It’s the best kind of challenge that fires up your brain like nothing else.
To integrate is to take up arms, embrace challenge, stake one’s soul and take on a enemy that one can never be sure is vanquishable. There is no satisfaction quite like that of finally conquering an especially challenging integral after days of struggling.
Have you even math-d until you’ve integrated?
- c