Archive for October, 2012

Why, why, why?Image

Since Halloween is coming soon, I will answer the question.

Proof of Halloween = Christmas

Halloween = 31 Oct = 3×8+1 = 25 Dec = Christmas 🙂

As we all know integration doesn’t always have a closed form solution therefore it is harder than differentiation in general.  So, solving an integration problem sometimes require a bit of creativity and luck.  Last week I saw a post on a forum.  The question is to evaluate the following definite integral,


What grabs my attention is the guy who posts this said his teacher can’t do it, then not long after someone replies the poster a solution with closed form answer for the indefinite integral with partial fraction.  But he didn’t bother to plug the upper and lower limits back in.  I guess it’s too tedious to do.


Partial fractions yields,




Then plug in the upper limit and lower limit and after 5 minutes of boring calculation, the answer would be \displaystyle\frac{\pi}{12}

This whole process can take up 15 minutes to an hour to do depending on a person’s math skill.  If you don’t believe me, you can try it on a piece of paper.

But there is a good news, you can solve this in a minute!  This technique isn’t often taught in many schools.  Maybe because not many people have known it or seen it before.  Anyway here it is,

Let \displaystyle x=\frac{1}{w}, hence \displaystyle dx=-\frac{1}{w^2}dw

Then we have,

\displaystyle I=\int_{1/\sqrt{3}}^{\sqrt{3}}\frac{dx}{(x^3+1)(x^2+1)}=\int_{1/\sqrt{3}}^{\sqrt{3}}\frac{w^3\;dw}{(w^3+1)(w^2+1)}

Since w is just a dummy variable in the definite integral.  We have,

\displaystyle I=\int_{1/\sqrt{3}}^{\sqrt{3}}\frac{dx}{(x^3+1)(x^2+1)}=\int_{1/\sqrt{3}}^{\sqrt{3}}\frac{x^3\;dx}{(x^3+1)(x^2+1)}

Now we can add them together as following,

\displaystyle 2I=\int_{1/\sqrt{3}}^{\sqrt{3}}\frac{dx}{(x^2+1)}=\tan^{-1}x\Bigg|_{1/\sqrt{3}}^{\sqrt{3}}=\frac{\pi}{3}-\frac{\pi}{6}=\frac{\pi}{6}


\displaystyle I=\frac{\pi}{12}

I have always wanted to have a powerful math software like Matlab or Maple on my iPhone.  I did have wolframalpha installed in my iPhone, but wolframalpha can’t handle basic programming like maple does.  In maple, you can write a for loop to generate a sequence and while loop to do iterative calculation.  Three weeks ago when I was casually browsing on youtube, I accidently bump into a video showing an iOS app called “MathStudio”.  It looks promising, it has a functionality similar to Maple.  you can do symbolic manipulation like factoring polynomials, solving equations (even ODE), integration, summation, graphing, and more.  I really like the interface.  To check out the demo of the app, you can click the link  The app isn’t free.  It costs $19.99 USD.  It is a nice investment if you need to do more than a highschool arithmetic on the road quickly with an iPad or an iPhone.

what’s new from Maplesoft?

Posted: October 1, 2012 in Mathematics

Maple 16Similar to Matlab, Mathematica, etc.  Maple is a math software that can do more than just spitting out numeric results, it can manipulate symbolic calculation!  The newest version up-to-date is Maple 16.01.  You can see almost an hour long “what’s new” video here  I recommend you to take a look at it if you are a student or a teacher and decide whether to install it in your machine to play around with it.