integration - where to start

I've just been trying to do some indefinite integration (I have done up to but not beyond C4). In the end I gave up and went on to a web page that did it with steps - it turned out to be much easier than I had expected.

My question is, how do these online integrators make decisions? Does anyone on TSR have an algorithm/method that they use to decide what to do that they would like to share?

Reply 1

Zacken

22

Original post by maggiehodgson

I've just been trying to do some indefinite integration (I have done up to but not beyond C4). In the end I gave up and went on to a web page that did it with steps - it turned out to be much easier than I had expected.

My question is, how do these online integrators make decisions? Does anyone on TSR have an algorithm/method that they use to decide what to do that they would like to share?

It's more intuition that you gain from repeated practice, you recognise even/odd functions, what substitutions to use, whether IBP works, etc...

Basically, integration, imo, is more of an art that you perfect by practice than using an algorithm/method.

Reply 2

maggiehodgson

OP

14

Original post by Zacken

It's more intuition that you gain from repeated practice, you recognise even/odd functions, what substitutions to use, whether IBP works, etc...

Basically, integration, imo, is more of an art that you perfect by practice than using an algorithm/method.

Mmmm. I do agree that practice helps BUT these online algorithms don't have intuition. There must be a something somewhere.

If all else fails I'll revisit C3 and C4 modules.

Thanks

Reply 3

Zacken

22

Original post by maggiehodgson

Mmmm. I do agree that practice helps BUT these online algorithms don't have intuition. There must be a something somewhere.

If all else fails I'll revisit C3 and C4 modules.

Thanks

Oh, no, of course they've got algorithms - but as a human, you'd be better off developing intuition and experience than trying to memorise the hundreds of complex algorithms that would be much more work than just practise.

Reply 4

DFranklin

18

Original post by maggiehodgson

Mmmm. I do agree that practice helps BUT these online algorithms don't have intuition. There must be a something somewhat.

http://stackoverflow.com/questions/3239336/how-to-calculate-indefinite-integral-programmatically

Needless to say, this is not a good approach for someone who's just done C3/C4!

Reply 5

maggiehodgson

OP

14

Original post by DFranklin

http://stackoverflow.com/questions/3239336/how-to-calculate-indefinite-integral-programmatically

Needless to say, this is not a good approach for someone who's just done C3/C4!

Followed the link. whoa.

Thanks

Reply 6

ian.slater

12

Original post by maggiehodgson

My question is, how do these online integrators make decisions? Does anyone on TSR have an algorithm/method that they use to decide what to do that they would like to share?

The thing with A level questions is that you know you can solve them using only a short list of possible techniques. So you need to become familiar with the range of possible questions and how to crack them. In a way, symbolic integration software must work the same way ... you need a list of templates and you compare the question with the templates. But knowing in detail how they work won't help you ... they know stuff you don't need to.

Reply 7

maggiehodgson

OP

14

Original post by ian.slater

The thing with A level questions is that you know you can solve them using only a short list of possible techniques. So you need to become familiar with the range of possible questions and how to crack them. In a way, symbolic integration software must work the same way ... you need a list of templates and you compare the question with the templates. But knowing in detail how they work won't help you ... they know stuff you don't need to.

Thanks

That's a very interesting idea. Yes, I can imagine it working a bit like a compiler. It will parse a line and decide what's there.

I'm doing M3 after a long break since C4 and integration skills are again needed. Having nearly got to the end of chapter 1 I am beginning to recall past techniques and I'm sure it will all come back. I was just interested in how online calculators might work.

Your suggestion has probably satisfied my curiosity.
Thanks.

Reply 8

dpm

13

Original post by maggiehodgson

I've just been trying to do some indefinite integration (I have done up to but not beyond C4). In the end I gave up and went on to a web page that did it with steps - it turned out to be much easier than I had expected.

My question is, how do these online integrators make decisions? Does anyone on TSR have an algorithm/method that they use to decide what to do that they would like to share?

If we are only going up to the end of C4 then I tend to
(i) look out for anything that contains a function and some form of its derivative ..

i.e. g'(x).f(g(x)) or g'(x)/f(g(x)) if It is there then a substitution tends to work (or reverse chain rule if you prefer)

(ii) Failing 1 then I tend to check if we have the product of 2 fairly unrelated functions ... if so then parts is often the way to go

(iii) If we have a rational expression in which the denominator is "more complicated" than the numerator then a quick substitution may work ..

(iv) I also tend to look out for common "tricks" e.g. ln(x) (parts with 1.ln(x)) or "cyclic" situations (e.g. e^x. sin(x) ... parts twice with a re-arrangement and division by 2)

Reply 9

maggiehodgson

OP

14

Yes. All these tricks are coming back now, slowly but coming, but it's good to have been given a reminder. The one about the denominator has come in very handy for the question I'm just tackling.

Thanks

integration - where to start

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