Because integration isn't reverse differentiation, integration is area. And to find an area, the function has to be well-behaved (technically, integrable) across an interval. So if we know its value at a point, to know it at any other point we need it to be integrable on the interval between them.

Not played it myself, but have been reliably informed that it is The Traitors done right and is well worth playing.

I was also told to start with the simpler versions and work up to the more complex.

It's not a convention. It's a necessity.

In my head I'd be muttering "it's the zeroth de Rham cohomology of the space, what do you expect?". But only in my head.

is like Gandalf standing on the bridge declaring "you shall not pass".

Less poetically, as the ode is not defined we cannot use it to transfer information from one section to another.

The differential equation *itself* is not defined at x=0. So there is no way to bridge the gap between positive and negative x.

In terms for A level, I might say that an ODE allows us to use information about "where I am" and "how fast I'm going" to predict the future. But that x=0 singularity

But then if t = tan(u/2) and x = tan(u), then x = 2t/(1-t²) so although our substitution is trig-inspired, we don't actually need the trig part.

It's still not an obvious substitution, but I like how the argument goes once it's been made.

I then remembered that the standard trick to switch the parity of sines and cosiness is to use double angles. But I already had sec(u) so actually I wanted the half angle formulae. Then with t = tan(u/2) we have sec(u) = (1+t²)/(1-t²).

Amazingly, this works. I won't spoil the algebra.

Anyway, back to sec(u). I tried a few things - parts and identities - but kept going round in circles (ha ha). The problem was the usual one - it's easy to get some sin(u)'s, but only as even powers.

Incidentally, another student in the same class had asked about integrating sec(x), which is in the formula book so I'd taken the opportunity to talk about how we find a lot of integrals by differentiating lots of stuff and seeing what we get.

Is this available as an actual book? I can see links to slides based on the chapters, but not chapters themselves.

Tempting ... but fortunately, the person for whom I'm asking has access to a university library so I would hope they can borrow it for free.

"Buy used £151.95"!

what is ironing if not differential topology

In Britain we call this "a conference".