How to become good at integration? (1 Viewer)

[alexman123]

I've lurked here for a while, and I see many people with exceptional integration abilities, and I have no idea how they do it. I'm at the level where I can do basically any 4U integration question that I've found in past papers, but looking at much of the integrals in the marathons I have absolutely no idea how to reach the ability of some people (e.g. Paradoxica). Basically, what resources do these people use, where can I find large quantities of challenging yet doable integrals for some just above the standard 4U level of integration, and what resources are there on advanced integration techniques that we do not learn in the HSC? Thank you all in advance, it's highly appreciated.

 

[captainhelium]

Have you tried the 100 Coroneos integrals?

Most of them are pretty standard integrals but some can be challenging.

Also, maybe try the Bored of Studies trials lol they're ridiculous (e.g. Q11a) for this year's one - yes I know I can't even do the first short answer question lol)

 

[alexman123]

Have you tried the 100 Coroneos integrals?

Most of them are pretty standard integrals but some can be challenging.

Also, maybe try the Bored of Studies trials lol they're ridiculous (e.g. Q11a) for this year's one - yes I know I can't even do the first short answer question lol)

Oh yeah I've tried the Coroneos integrals, but idk I didn't really find any of them all that challenging. But yeah I saw that paper, and those are the types of integrals I'm looking for. But I don't know where to find many other similar integrals outside of those papers which is disappointing. Thanks though.

 

[andrew12678]

I've lurked here for a while, and I see many people with exceptional integration abilities, and I have no idea how they do it. I'm at the level where I can do basically any 4U integration question that I've found in past papers, but looking at much of the integrals in the marathons I have absolutely no idea how to reach the ability of some people (e.g. Paradoxica). Basically, what resources do these people use, where can I find large quantities of challenging yet doable integrals for some just above the standard 4U level of integration, and what resources are there on advanced integration techniques that we do not learn in the HSC? Thank you all in advance, it's highly appreciated.

I reckon Integration is a topic honed with practice, at first there may seem like a billion formulas to memorise but you'll eventually get the hang of it with enough practice. I don't think there's any particular exceptional resource, any of the good 4U resources are more than fine (eg. Terry Lee etc)

 

[pikachu975]

I reckon Integration is a topic honed with practice, at first there may seem like a billion formulas to memorise but you'll eventually get the hang of it with enough practice. I don't think there's any particular exceptional resource, any of the good 4U resources are more than fine (eg. Terry Lee etc)

I think the OP said they're good with 4U integration but they want to be good at the next level ones in the integration marathon.

 

[KingOfActing]

Practice makes perfect

 

[alexman123]

Practice makes perfect

The problem is I'd like a lot of resources to practice with. There seems to be a gulf between normal 4U integration (I've never come across a HSC integral which has posed a problem for me) and the higher level integration I come across, so I'd like some resources to bridge the gap and get me to that level.

 

[KingOfActing]

The problem is I'd like a lot of resources to practice with. There seems to be a gulf between normal 4U integration (I've never come across a HSC integral which has posed a problem for me) and the higher level integration I come across, so I'd like some resources to bridge the gap and get me to that level.

There probably isn't one. The reason is that there is no "in-between" techniques. You just have to tackle the harder integrals until you get good at them too.

 

[seanieg89]

So this answer assumes that you are just interested in getting better at integration itself, rather than HSC integration in particular. If you are already on top of the integrals in past papers and the standard HSC textbooks, you will be absolutely fine for HSC integration. If you want to improve your integration in general, then there are many sources. (But do not use any higher level techniques you learn from these books in HSC examinations.)

Some books you might find of use if you haven't checked them out already:

Inside interesting integrals - Nahin
Irresistable integrals - Boros,Moll

You will almost certainly learn some new tricks from these books, and get some nice integrals to practice them on. Keep in mind that you might need some mathematical maturity beyond the high school level for certain topics. The Nahin book is newer, and its introduction references the differences in style/philosophy with the Boros/Moll classic. You might find one more to your taste than the other, but the content is not identical so check out both.

The handbook of integration - Zwillinger

Kinda different in style, and maybe less useful for learning than the above two. More a book to have on your desk to have a checklist of things to try when it comes to integration etc. In other words, a better reference book than a textbook.

A treatise on the integral calculus - Edwards

Quite old fashioned, but lots of problems you can mine from here on integration and applications of calculus etc.

Once you have mastered some new techniques, you can find new problems to practice them on at math.stackexchange (questions with tags related to integration), brilliant.org (past integration contests), artofproblemsolving maybe as well etc. If you are driven I am sure you will find many more.

 

[alexman123]

That's exactly what I was looking for seanieg89!! Thank you so much!

 

[omegadot]

In addition to those texts listed by seanieg89 I would add:

1. The Soviet text Problems in Mathematical Analysis edited by Boris Demidovich has many problems using the various techniques of integration to solve. A few of the techniques Demidovich gives are not usually found elsewhere.

2. As seanieg89 mentioned, Joseph Edwards’ A Treatise on the Integral Calculus (Volume 1) is a particularly valuable source for many interesting integrals, and being published before 1923, is out of copyright meaning it can be readily found online.

3. Michael Spivak’s Calculus contains many interesting questions that use integration, like the proof that pi is irrational for example.

4. While it is a bit old, G. H. Hardy’s A Course of Pure Mathematics, is also a useful sources for questions, and being published before 1923 means it too is no longer under copyright.

 

[seanieg89]

That's exactly what I was looking for seanieg89!! Thank you so much!

No worries at all man, hope they are helpful for your particular interests .

 

[seanieg89]

Hadn't come across the Demidovich book before omegadot, thanks for that!

(Good to have lots of sources to mine for assignment/exam questions for my calculus students haha. Some of these older texts are great for this purpose.)