Integrals You Can Only Experience Once

There are easy integrals. There are hard integrals. There are simple integrals, nice integrals, ugly integrals, and horror integrals. But there are also integrals so exquisitely beautiful, that you can only experience their beauty once in your lifetime. Like a firework they come, a burst of colour and vibrance, dazzling, yet ephemeral.

Once you have seen them, their charm and brilliance can fade away; once you know the solution, you know how to solve all integrals of that form, and it melds into but another tool in your integration arsenal. But that initial moment of realisation when you experience the solution for the first time – when your eyes are opened to something entirely new – that, is truly special.

$e^x \sin{x}$

You’re often shown this soon after being introduced to Integration by Parts.

$\ln{x}$

Ah yes, more parts. This one’s so special I wrote an article dedicated to just it – see Integrating the Logarithm.

$\sin(\sqrt{x})$

This integral pretty perfectly illustrates why some integrals you can only experience once. The first time you try to integrate it, it can be pretty daunting. You might try to substitute $\sqrt{x}$, in which case you find

And hey, that’s integration by parts!

But soon you realise $\sqrt{x}$ is almost always safe to substitute, and integrals like this become trivial.

$\sec{x}$

Oh yes, another special one. See Integrating the Secant.