HSC 2012-14 MX2 Integration Marathon (archive) (2 Viewers)

[Sy123]

Re: MX2 Integration Marathon

 

[FrankXie]

Re: MX2 Integration Marathon

I didn't forget, I just didn't bother. Any rational function can just be done by partial fractions, which isn't terribly exciting.

ok I reckon it is not a good question coz I hate partial fractions too. I wrote this question was to try using the technique

which reduce the integral into via cancelling common factor, and via substitution

but unfortunately partial fraction is still needed to solve these two

 

[FrankXie]

Re: MX2 Integration Marathon

hah, this is my favorite one. use substitution



then notice that



It follows that



which gives

 

[dan964]

Re: MX2 Integration Marathon

Dont forget to try this one

answer has been checked with Wolfram
View attachment integral.pdf
edit: there is an error in the second last line. One of the pluses should be a minus. meh, who cares.

 

[Carrotsticks]

Re: MX2 Integration Marathon

I would bet all my money, my house, my dog and my car that there is no elementary primitive.
Just saying.

 

[dan964]

Re: MX2 Integration Marathon

I would bet all my money, my house, my dog and my car that there is no elementary primitive.
Just saying.

which integral?

 

[Carrotsticks]

Re: MX2 Integration Marathon

 

[Trebla]

Re: MX2 Integration Marathon

I would bet all my money, my house, my dog and my car that there is no elementary primitive.
Just saying.

Can you formally show that there is no elementary primitive?

 

[Carrotsticks]

Re: MX2 Integration Marathon

Can you formally show that there is no elementary primitive?

Yep, you most certainly can.

Whilst I'm aware of this theorem and the basic gist of it, I confess that I haven't had the opportunity to really dig into the meat of it.

 

[Sy123]

Re: MX2 Integration Marathon

Can you formally show that there is no elementary primitive?

Yep, you most certainly can.

Whilst I'm aware of this theorem and the basic gist of it, I confess that I haven't had the opportunity to really dig into the meat of it.

Is there a rigorous definition of "elementary function"?

What makes the gamma function (for example) not elementary, but something like ln() or sin() elementary even though they are both transcendental?

 

[seanieg89]

Re: MX2 Integration Marathon

Is there a rigorous definition of "elementary function"?

What makes the gamma function (for example) not elementary, but something like ln() or sin() elementary even though they are both transcendental?

I would say something like

Elementary function = anything formed recursively (using the operations of +,-,*,/,^ and composition) from: the reals, the monomial x, the 3 trig funcs and their inverses, the 3 hyp trig funcs and their inverses, the exponential and the logarithm.

I don't know that theres any universally agreed upon definition though, its not particularly important because its just a matter of conventions.

 

[Carrotsticks]

Re: MX2 Integration Marathon

The standard definition is a function that can be constructed via a finite number of applications of roots (of degree n, tbh I don't remember if n had to be rational), exponentiations and logarithms under your elementary operations (hence the name) and compositions. With enough playing around (using the fact that we can construct the trigs using complex exponentials), then we've also got the trigs and inverse trigs thrown in there too. The gamma function isn't considered elementary I presume it cannot be constructed using a finite number of 'allowed' things I've listed. As for the proof of that, I am not too sure to be honest. I've not yet had a proper look into that.

EDIT: Beaten.

 

[Carrotsticks]

Re: MX2 Integration Marathon

Thinking about it a bit more carefully now, the question "What makes a function elementary" is a lot more complicated than at first glance. I'm even getting a little confused regarding the definition I've learned (finite applications of etc). Asking myself questions like is an integral constructed via an infinite sum elementary? Is a function produced using the infinite geometric sum not elementary then? How about strange numbers like Liouville's Constant? Roots of polynomials of degree 5 or more?

 

[seanieg89]

Re: MX2 Integration Marathon

I'm pretty sure all constant functions are considered elementary, no matter how unusual the number. (Most numbers are transcendental anyway, which imo is weirder than being a root of a poly that cant be expressed by radicals).

Also, the recursive building of an elementary function from the "building blocks" has to be finite. Then you can study this thing as an algebraic object using Galois theory. (Which is how you prove that certain things don't have elementary primitives).

So no, infinite sums/products don't count. Otherwise we are opening the gates for pretty much all the special functions.

 

[Carrotsticks]

Re: MX2 Integration Marathon

I'm pretty sure all constant functions are considered elementary, no matter how unusual the number. (Most numbers are transcendental anyway, which imo is weirder than being a root of a poly that cant be expressed by radicals).

Also, the recursive building of an elementary function from the "building blocks" has to be finite. Then you can study this thing as an algebraic object using Galois theory. (Which is how you prove that certain things don't have elementary primitives).

So no, infinite sums/products don't count. Otherwise we are opening the gates for pretty much all the special functions.

Where I am not sure is when you consider things like Taylor Series for say the sine function. The sine function is most certainly elementary yet the Taylor Series is of course infinite, so it cannot be elementary?? This is where I'm not completely sure.

 

[seanieg89]

Re: MX2 Integration Marathon

Where I am not sure is when you consider things like Taylor Series for say the sine function. The sine function is most certainly elementary yet the Taylor Series is of course infinite, so it cannot be elementary?? This is where I'm not completely sure.

Well the precise defn would be: CAN be obtained by a finite recursive application of the allowed operations on the listed basic functions.

sin(x) can be obtained by applying absolutely 0 operations as it is itself one of the basic functions. Hence it is elementary.

The word "can" is key. The fact that it can also be written as an everywhere convergent infinite series as well is irrelevant. Otherwise the function 1/(1-x) on |x|<1 would have to be excluded as well, as it is the sum of the geometric series.

In fact all of the elementary functions are nice enough to be analytic on their appropriate domains I think.

 

[Carrotsticks]

Re: MX2 Integration Marathon

Well the precise defn would be: CAN be obtained by a finite recursive application of the allowed operations on the listed basic functions.

sin(x) can be obtained by applying absolutely 0 operations as it is itself one of the basic functions. Hence it is elementary.

The word "can" is key. The fact that it can also be written as an everywhere convergent infinite series as well is irrelevant. Otherwise the function 1/(1-x) on |x<1| would have to be excluded as well, as it is the sum of the geometric series.

Cheers, that's what I was looking for.

Funny you mention that specific geometric sum, I was thinking precisely that to put in my previous post, but decided to use the Taylor Series example to avoid any potential for the radius of convergence playing some role somehow.

 

[seanieg89]

Re: MX2 Integration Marathon

Cheers, that's what I was looking for.

Funny you mention that specific geometric sum, I was thinking precisely that to put in my previous post, but decided to use the Taylor Series example to avoid any potential for the radius of convergence playing some role somehow.

Its a pretty nice and simple sum lol.

 

[Sy123]

Re: MX2 Integration Marathon

I would say something like

Elementary function = anything formed recursively (using the operations of +,-,*,/,^ and composition) from: the reals, the monomial x, the 3 trig funcs and their inverses, the 3 hyp trig funcs and their inverses, the exponential and the logarithm.

I don't know that theres any universally agreed upon definition though, its not particularly important because its just a matter of conventions.

Well the precise defn would be: CAN be obtained by a finite recursive application of the allowed operations on the listed basic functions.

sin(x) can be obtained by applying absolutely 0 operations as it is itself one of the basic functions. Hence it is elementary.

The word "can" is key. The fact that it can also be written as an everywhere convergent infinite series as well is irrelevant. Otherwise the function 1/(1-x) on |x|<1 would have to be excluded as well, as it is the sum of the geometric series.

In fact all of the elementary functions are nice enough to be analytic on their appropriate domains I think.

Is there a reason for that specific list of functions to be considered elementary?

What if I say

Elementary = anything formed recursively (using the operations of +,-,*,/,^ and composition) from: the reals, the monomial x, the 3 trig funcs and their inverses, the 3 hyp trig funcs and their inverses, the exponential and the logarithm, AND the Gamma function

Is there a reason why my this addition to the definition is wrong? Is the concept of elementary functions purely a categorization of mathematicians?

 

[dan964]

Re: MX2 Integration Marathon

yeah there isn't.