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,

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

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.

\displaystyle\int\frac{dx}{(x^3+1)(x^2+1)}

Partial fractions yields,

\displaystyle=\int-\frac{2x-1}{3(x^2-x+1)}+\frac{x+1}{2(x^2+1)}+\frac{1}{6(x+1)}\;dx

\displaystyle=-\int\frac{d(x^2-x+1)}{3(x^2-x+1)}+\int\frac{d(x^2+1)}{4(x^2+1)}+\int\frac{dx}{2(x^2+1)}+\int\frac{d(x+1)}{6(x+1)}

\displaystyle=-\frac{\log(x^2-x+1)}{3}+\frac{\log(x^2+1)}{4}+\frac{\tan^{-1}x}{2}+\frac{\log|x+1|}{6}+C

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}

Therefore,

\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 http://www.youtube.com/watch?v=XebH_p7reTk.  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 http://www.maplesoft.com/demo/streaming/m16WhatsNew.aspx.  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.