Integration is one of, if not the most, sophisticated of mathematical pleasures. It’s an enthralling dance of ingenious problem-solving, daring adventure and mathematical warfare. It’s an uncountably infinite lifelong journey of learning, upon which we are forever taking finite steps. It is, quite simply, so much fun.
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 mathematical landscapes awaiting exploration. The rays of dawn 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 too far off; you’ve just got a couple more symbols to worry about. If you’re into hyper-realistic real-time 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 seven oceans, with tides that shape the shore of your understanding, 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, in the uncountably infinite number of potential problems, this still leaves more than enough to satiate our mathematical thirst – but the imposing enormity of the expanse that is integration is quite humbling to appreciate; no doubt, it is not dissimilar from realising one’s insignificance within the universe.
Integration trains you in so much more than simply memorising antiderivatives. It trains you to be swift but accurate in your algebra. With enough time, integration becomes second nature – and that is when the really juicy stuff comes.
And of course, integration is by no means restricted to pure maths. It’s behind probability in statistics, polarisation and fields in physics, and spectra analysis in chemistry.
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, lush, rich and full.
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 conquering an especially challenging integral with impeccable flourish.
Have you even math-d until you’ve integrated?
- c