It's not too hard to do either: if you remember down there when I differentiated X^2 and ended up with 2x you can do a reverse operation and it turns out the formula, yielded by some relatively straightforward arseing about with algebra or geometry, to integrate a simple function of the form
n
ax
is
a 1 n+1
--- x
n+1
which happily yields x^2
However the devil in the detail is that the result of integration needs to have an arbitrary constant added on as differentiating x^2 + a (where a is any constant) gives 2x so the reverse operation has to add this "constant of integration" back in. Apparently this makes these "Indefinite Integrals" but luckily for me you can get them to be a bit more definite when you subtract one from the other.
And subtracting the results of integrating a function for different values of the dependent variable is the one bit of calculus I vaguely remember being shown at school (I never did A-levels as I loathed the two maths teachers with a passion so I'm guessing this got touched on somewhere in AO level1 pure maths) probably for finding areas under curves.
Oh and I got a book on Calculus as well (see - still a swotty dragon, reading around the course now!) called "Calculus made Simple (for scaly green firebreathing creatures like you)2" written by some guy in the early 1900's and apparently still the definitive text on the subject today. I will let you know how I get on
1 This was an old UK exam that I think got dropped sometime in the 80's. I, being a swotty hatchling, was in an "advanced" stream at secondary school and we took our O levels (called GCSEs now) a year early at 15 and we also did a couple of these "midway between an O level and A level". I had a google and although I couldn't find a AO level paper I did come across a marking scheme for one here: www.edexcel-international.org/VirtualContent/49349/7362_PURE_MATHEMATICS_F.pdf and it does have a little bit of calculus in it.
2 I think I might have imagined the subtitle.
