A longer look at the Si(x) function

Sinx/x can’t be integrated into an elementary function – instead we define:

Screen Shot 2016-05-21 at 9.31.12 PM

Where Si(x) is a special function.  This may sound strange – but we already come across another similar case with the integral of 1/x.  In this case we define the integral of 1/x as ln(x).  ln(x) is a function with its own graph and I can use it to work out definite integrals of 1/x.  For example the integral of 1/x from 1 to 5 will be ln(5) – ln(1) = ln(5).

The graph of Si(x) looks like this:

Screen Shot 2016-05-21 at 9.45.57 PM

Or, on a larger scale:

Screen Shot 2016-05-21 at 9.46.04 PM

You can see that it is symmetrical about the y axis, has an oscillating motion and as x gets large approaches a limit.  In fact this limit is pi/2.

Because Si(0) = 0,  you can write the following integrals as:

Screen Shot 2016-05-21 at 9.30.25 PM

How to integrate sinx/x ?

It’s all very well to define a new function – and say that this is the integral of sinx/x – but how was this function generated in the first place?

Well, one way to integrate difficult functions is to use Taylor and Maclaurin expansions.  For example the Maclaurin expansion of sinx/x for values near x=0 is:

Screen Shot 2016-05-21 at 9.31.00 PM

This means that in the domain close to x = 0, the function sinx/x behaves in a similar way to the polynomial above.  The last part of this expression O( )  just means everything else in this expansion will be x^6 or greater.

Graph of sinx/x

Screen Shot 2016-05-21 at 9.28.31 PM

Graph of 1 – x^2/6 + x^4/120

Screen Shot 2016-05-22 at 6.38.28 AM

In the region close to x=0 these functions behave in a very similar manner (this would be easier to see with similar scales so let’s look on a GDC):

So for the region above (x between 0 and 2) the 2 graphs are virtually indistinguishable.

Therefore if we want to integrate sinx/x for values close to 0 we can just integrate our new function 1 – x^2/6 + x^4/120 and get a good approximation.

Let’s try how accurate this is.  We can use Wolfram Alpha to tell us that:

Screen Shot 2016-05-22 at 6.49.08 AM

and let’s use Wolfram to work out the integral as well:

Screen Shot 2016-05-22 at 6.49.59 AM

Our approximation is accurate to 3 dp, 1.371 in both cases.  If we wanted greater accuracy we would simply use more terms in the Maclaurin expansion.

So, by using the Maclaurin expansion for terms near x = 0 and the Taylor expansion for terms near x = a we can build up information as to the values of the Si(x) function.

Essential resources for IB students:

1) Revision Village

Screen Shot 2021-05-19 at 9.55.51 AM

Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams.  I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

Screen Shot 2018-03-19 at 4.42.05 PM.png

There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank of graded questions.  What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

Screen Shot 2021-05-19 at 10.05.18 AM

The Practice Exams section takes you to a large number of ready made quizzes, exams and predicted papers.   These all have worked solutions and allow you to focus on specific topics or start general revision.  This also has some excellent challenging questions for those students aiming for 6s and 7s.

Each course also has a dedicated video tutorial section which provides 5-15 minute tutorial videos on every single syllabus part – handily sorted into topic categories.

2) Exploration Guides and Paper 3 Resources

Screen Shot 2021-05-19 at 6.32.13 PM

I’ve put together four comprehensive pdf guides to help students prepare for their exploration coursework and Paper 3 investigations. The exploration guides talk through the marking criteria, common student mistakes, excellent ideas for explorations, technology advice, modeling methods and a variety of statistical techniques with detailed explanations. I’ve also made 17 full investigation questions which are also excellent starting points for explorations.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.