Ho ho ho. Mr. Barley has done it again. Provided an interesting maths lesson? Yes. Made me laugh? Yes. But this time we got the added bonus of the (hopefully to be patented) ‘Barley Method’ ^_^. While I’m quite sure this method isn’t absolutely original, his deliverance and confidence can’t be faulted on the idea.
Mathematics Integration, Pure 3. The author’s example in the EdExcel textbook consists of a notoriously foul page of formulae, of which he suggests memorising. A face contorted in hearty disbelief and ridicule, we simply get to hear one of the now expected book-bashings as a result of Mr Barley’s supreme mastery of mathematics. Mr Barley’s method, in comparison, consists of ‘performing Integration by guessing’. “Foolproof” he says, with a chuckle and a grin. He’s got gusto. Yes, no doubt at all. I can’t help but laugh as he nonchalantly begins, pen poised on the whiteboard, to explain what he himself credits as ‘The Barley Method’. Genius ^_^. We learn that (after re-capping the ‘chain rule’ for composite functions), Integration is simply the inverse process of Differentiation. (No way! Really?!) So what better way to do Integration that to avoid doing it altogether? We’ll just stick will good old differentiation thanks.
Okay. From here on out the entry gets really mathematical, so I apologise for scaring anyone with what will not be a regular occurrence, I can assure you. :) We take an educated guess at what the integral would be based on the mathematical rules of certain functions, then differentiate that function to get an answer that is close to the original. Off by a certain numerical factor. So all we then do is apply what would negate that effect to the original equation. Thus, if we get our differential to be 30 times bigger than our given equation, we simply divide the original by 30.
> Basically we differentiate the power outside the bracket, then differentiate the bracket itself and multiply the function by the result.
Integrate: (6x+1)^4 dx
To integrate we increase the power by one and divide by the new number, but since this function is composite, we can only say for sure to increase the power and follow the chain rule.
Now differentiate – multiply by the power, drop the power down by one: 5(6x+1)^4
And, using the chain rule for the bracket: (Chain rule being dy/dx = dy/dt x dt/dx)(Assuming we let t=6x+1, then dt/dx=6)
We get 6 as the differential of the bracket. Therefore, our differentiating is done and we have this answer:
5(6x+1)^4 x 6
So we know from our answer that we are 30 out from the original equation (6×5). Therefore the integral of (6x+1)^4 dx is:
1/30 x (6x+1)^5 + C
And thus concludes the Barley Method. I appreciate those not doing Maths will probably be puzzled and/or mortified by the sight of that, and I’ve just realised how geeky I probably sound by writing about Maths :P. Still, it’s good revision.