Integration in Maths: The Paddington Bear Approach

Integration and differentiation are a huge part of A Level maths.

A quick count of the marks in a recent (November 2020) A Level paper showed that 99 out of the 200 marks came from them. That’s nearly 50% of all the marks. Now whilst most pupils can cope with differentiation, nearly everyone finds integration in maths tricky. But why?

What’s the Difference?

The big difference is that differentiation is really the starting point and there are fixed ways of doing it. If you have a function in a particular form there is really only one way to differentiate it. These are nicely learnable. A product uses the product rule, a compound function uses the chain rule etc… There are even nice easy ways to remember them.

Product rule: “Keep Differentiate + Differentiate Keep” what could be simpler?

Integration, on the other hand, has lots of different methods and, whilst each method on its own is very learnable and not at all difficult, not one of them ALWAYS works – there is no silver bullet.


This wouldn’t, on its own, be a problem.

After all, just learning a few different methods isn’t the end of the world. The issue is that spotting which method to use can be very hard. Two very similar looking functions can need very different ways of approaching them. Changing  to , just by sticking in an innocent looking little “squared” changes it from a very simple, just write it down, integration problem to one that requires some serious university level maths.

Integration in maths

To make it worse, there can often be more than one method that does work. This sounds great, but the different methods can lead to different forms of the answer achieved. Many an A Level pupil will have spent some considerable time trying to work out where they have gone wrong when their answer doesn’t match the one in the back of the book or the “show that” in the question, only to discover that their answer was correct, just written in a different way.



So how can an A Level pupil “get there”?

The main thing, as in any A Level maths question, is to remember that you are doing an A Level maths question. This means that it will definitely work and will only need the methods that

you have been taught. So DON’T PANIC!

Integration in maths

Next, start the question by employing the Paddington Method – give it a really hard stare!  Yes, seriously….

If you look really hard at the function and think “if I differentiated something and ended up here, what must I have started with?” this can often get you most of the way there.

You are used to doing this. No one, probably(!), actually divides numbers – if asked “What is 42 divided by7?” we all think “what do I have to multiply 7 by to get 42”

So can you see where you must have started from?

  • Is there a “base function” that must have been involved? Sin must have come from cos, ex stays the same etc..
  • Might I have used the chain rule to get this product of two functions, or maybe the product rule produced this sum.
  • Is it a fraction where the top is the differential of the bottom, or the differential of the inside of the brackets of the bottom?
  • If the answer to the above is yes then you have almost certainly done most of the work already and just need to think about the multipliers..

Now I am not saying that this method is easy, but it is very useful and it can be practiced. You can get a set of ones that work like this from here and work your way through, seeing how many you can do in 5 minutes. Keep coming back to it and trying again so that you get better. The time spent on this now can save you lots of time and grief in an actual exam.



But what if even the hardest Paddington stare doesn’t work?

You’ve eaten all the marmalade sandwiches, tried again and it still doesn’t work. Well, in this case the reason why it doesn’t work will tell you what to do next.

A product that didn’t come from chain rule will need differentiation by parts or a fraction that isn’t from ln or chain rule needs to be flipped with a substitution or separated with partial fractions.

And don’t forget, rule number 1 – it does work, you are doing A Level questions! It’s not like real life where there are times when we can’t solve things. So, if it seems to be going wrong and taking too long you have either made a mistake or are doing it the wrong way…

So the key to integration in maths is…

Practice your Paddington stare and then just apply some common sense.  And like all good Paddington stories, it will work out in the end…

Practice Paddington questions
                      Practice Paddington answers


Want your child to succeed? Why not try one of our expert tutors. You can read why this helps here: Why Should I book a Maths Tutor


Andy SA bit about Andy:

Andy is incredibly good at breaking down maths into easy to understand chunks, whether it be number bonds with primary pupils, the dreaded fractions with KS3 or the scarily hard Further Maths and undergraduate topics.

With a firm belief that you can teach anyone anything as long as it is approached in the right way, Andy has led a class of “sink” pupils in an inner city comprehensive through a lesson where they all “discovered” Pythagoras’ Theorem (and still remembered it 3 months later!) and has had year 8 pupils working through Year 13 algebraic proofs using multilink cubes.

You can enquire about tutoring with Andy here

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