Wikipedia:Reference desk/Archives/Mathematics/2011 April 1

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April 1

Difficult integration question

Hi,

How do I integrate sqrt(x^3+x^4)? I have tried almost every method that I know off (I have done IB Mathematics HL) but I still can't do it. Any help will be appreciated.

Thanks —Preceding unsigned comment added by 169.232.101.13 (talk) 04:39, 1 April 2011 (UTC)

This help? [1] Shadowjams (talk) 08:37, 1 April 2011 (UTC)

I tried Wolfram already but I want to know how to actually integrate it! —Preceding unsigned comment added by 169.232.101.13 (talk) 14:49, 1 April 2011 (UTC)

I haven't worked it through, but my initial impression is that one would want to first pull a factor of x^2 (or even x^3) out of the square root sign to simplify. I would then look at doing an Integration by substitution, probably by using the substitution u = sqrt(x). You may be lucky, and run across a form listed in Integration by reduction formulae, in which case you can apply the technique there, but if not, then I would probably play around with various Integration by parts schemes. WolframAlpha says the answer contains a inverse hyperbolic sine function, so at some point, you'll probably have to recognize that the derivative of the inverse hyperbolic sine is 1/sqrt(u^2 + 1) or that the derivative of the inverse hyperbolic sine of a square root is 1/(2 * sqrt(u) * sqrt(u+1)). Often integration boils down to rearranging the integral until it resembles one of the standard forms on a Table of integrals. -- 140.142.20.229 (talk) 21:10, 1 April 2011 (UTC)

Wait, hold on. I think I've got it. Do the u = sqrt(x) integration by substitution. Simplify by pulling out the common factor under the radical. Realize that the new form is listed at List of integrals of irrational functions. Then "Plug and Chug", as my math professor used to say. Then, if you need to further simplify/match with WolframAlpha, realize that ln(sqrt(x) + sqrt( x + 1)) = sinh^-1(sqrt(x)) (That last is a straight function evaluation, no derivatives/integration required.) -- 140.142.20.229 (talk) 21:18, 1 April 2011 (UTC)

How to choose substitution function while finding antiderivative by method of substitution

If we are interested in finding antiderivative of an arbitrary function by method of substitution then there is no general method to find the substitution function.When the mathematician or the instructor chooses a substitution function he generally does not give the reason for making the particular substitution.He does not divulge how he arrived at the particular idiosyncratic substitution or what thought led him to a unique substitution.Is this done by a heuristic choice from a set of infinite possible substitution functions.But heuristic approach does not guarantee to make the process of finding antiderivative simple.One may end up in a mess if he fails to find the particular unique substitution function.Thus I find the application of method of substitution a very unscientific grey area.Ican give an example to illustrate my point.Take for example Integral dx/{(p*x+q)Sqrt(a*x^2+b*x+c)}.As far as I know this indefinite integral can be simplified by the substitution (p*x+q)=1/z.But my question is how will one arrive at this substitution if he has no apriori hint. What is the reasoning and rationale for this choice in particular.Can somebody shed light on this topic and give answer in positive or negative.Ichgab (talk) 13:11, 1 April 2011 (UTC)

When I teach, I do tell my students how I pick the substitution. I don't do problems as complicated as you just gave though, usually it's picking u to be whatever is inside some other function. But, I always tell them it's basically just guessing and the more problems you do, the better at guessing you will get. The way someone knows to choose (px+q) = 1/z is to see that sort of problem before and then remember that such a thing might work. Or, see enough different examples to learn to experiment with what you choose. Or, to be extremely creative. StatisticsMan (talk) 19:50, 2 April 2011 (UTC)

I'd like to add that some substitutions often seem very artificial from the "inside the function" perspective, like trigonometric substitution and Weierstrass substitution. There are many applicable tricks and heuristics, but there seems to be no general method for cooking up useful substitutions. Does anyone know if the problem is decidable? Sławomir Biały (talk) 20:53, 2 April 2011 (UTC)

Solving by detour

Is there a general name for the method of solving a problem "by detour", so to speak, in a three-step process as follows: If we can't get from f to g directly (I'm using $f\rightsquigarrow g$ for "somehow getting from f to g), we use $f\rightharpoonup f'\rightsquigarrow g'\rightharpoondown g$, where $\rightharpoonup$ indicates some operation, such as a Fourier transform or a rotation of a Rubik's cube, and $\rightharpoondown$ its inverse. — Sebastian 17:10, 1 April 2011 (UTC)

In complexity theory this is called reduction. —Bkell (talk) 17:21, 1 April 2011 (UTC)

Conjugation, especially in case of the Rubik's cube (which is a group). – b_jonas 18:02, 1 April 2011 (UTC)

Thanks for both your answers. I love the example given in Inner automorphism: "take off shoes, take off socks, put on shoes". That seems to be as generally understandable as it gets! BTW, another example would be the use of logarithms to reduce multiplication to addition. — Sebastian 18:48, 1 April 2011 (UTC)

Functor? —Preceding unsigned comment added by 78.127.33.243 (talk) 21:12, 1 April 2011 (UTC)

That seems to be too general; intuitively at least, you can't have an inverse of $\rightharpoonup$ if it's forgetful. — Sebastian 07:33, 2 April 2011 (UTC)

But it doesn't have to be forgetful. This paper: http://arxiv.org/abs/0906.3330 explains how the Fourier transform can be views as an analytic functor. —Preceding unsigned comment added by 78.127.68.220 (talk) 11:57, 2 April 2011 (UTC)

Still more Fundamental Theorem of Algebra

So, the usual blurb: I'm considering the proof of the FTA that takes some function $f(z)$ and considers the mapping into the w plane given by $w=f(C_{r})$ where $C_{r}$ is the circle centred at the origin, radius r in the z plane.

I'm considering the behaviour in the w plane for functions that have roots whose multiplicities are of $3^{rd}$ order or higher and to get a real understanding of what's going on, I'm considering the differing behaviours for even and odd multiplicities. I picked $f(z)=[z-(3+4i)]^{n}$ and considered n=3, 4. For n=3, I was confused by the appearance of the second 'loop'; I was expecting it precisely when r=5, ie the modulus of the root, but it is quite apparent when r=4.8. Then, for n=4, the third 'loop' is very small but certainly present for r=4 and then it disappears for r=5! (That's an exclamation mark, not a factorial.) What is going on? Can someone please explain? Thanks. asyndeton talk 21:16, 1 April 2011 (UTC)

I'm sorry -- I think my earlier comments led you to expect a simpler pattern than the one that really appears. I could say it was in order to make sure you make your own observations, but I actually had it wrong myself. Now I have done some more careful sketching, and what I now think you ought to observe for n=3 is this: The image of the circle begins to overlap itself at r = 2.5√3 ≈ 4.33, but the two self-intersections that result are initially some distance from the origin. There is now a large loop clockwise around the origin, and a smaller one counterclockwise around the origin, with two overlapping segments connecting them (so there's no net winding around the origin). The small loop contracts around the origin as r goes towards 5. Exactly at r=5, it has shrunk to nothing, but for r>5 it reappears as a small clockwise loop, and the two connecting segments now form two clockwise half-loops so there are now three loops around the origin in total.

For n=4 I expect something similar, except that the overlap begins as early as r = 2.5√2 ≈ 3.54, and the configuration that shrinks to nothing around the origin for r=5 is more complex but again reappears in a mirrored version. In both cases you should be able to see that there are inflection points close to the origin for r just below 5, but none afterwards. One or more counterclockwise turns around the origin become clockwise (or vice versa), while the qualitative behavior further out doesn't change.

(And it seems this would indeed be a good time to attempt the winding number once again). –Henning Makholm (talk) 00:08, 2 April 2011 (UTC)

Math for Social Science

I'm a student of social sciences, so my knowledge in math is pretty much limited to some elementary education in statistics and what little I remember from school. Having a deep interest in topics like societal collapse and agricultural sustainability has led me to articles like these:[2] [3]

As you can see, after scrolling down a bit, you will find most of the arguments in Math. My question is how serious would it be for me to try and study math and reach the level of understanding this? Would it only require a particular field of math? I guess i am looking into knowing about how many years and what steps I would need to take right away.

At this point I have no idea if this would be something I could manage with a couple of introductory books or if it would take many years.

Thank you. Maziotis (talk) 23:35, 1 April 2011 (UTC)

From a cursory glance (and from a mathematical point of view) the mathematics here does not seem overly advanced. The authors do manage to make it look forbiddingly complex, possibly because of the number of different effects they model at the same time, but it looks like each individual piece of it is not conceptually difficult. It builds on basic calculus with some large system of coupled ordinary differential equations, some multivariate calculus for analysing their structure, and some general theory of dynamic systems (concepts of stability of solutions and so forth). These are, I think, the standard tools of the trade of macroeconomics in general, and you'll probably need to become conversant with them if you want to say something quantiative about collapsing societies.

Without knowing anything about your academic situation or location, I'd tentatively suggest a semester or two of calculus, plus see if you can find an introduction to economic modeling. –Henning Makholm (talk) 01:05, 2 April 2011 (UTC)

Before looking at calculus you might need to look at Precalculus and Elementary algebra. Once your fairly happy with calculus you would need to look at some Differential equations especially the Lotka–Volterra equation which is mentioned in the article. Mathematical economics gives an overview of some areas you could extend your study and Mathematical sociology and Computational sociology look at some of the more advanced and interesting reaches of the subject.

So it is quite an open ended question. A course like "mathematics for the arts/humanities" would be a must, but you could study the mathematical aspects in great depth and maybe specialising in it throughout your studies.--Salix (talk): 07:05, 2 April 2011 (UTC)

Thank you for the feedback. I am going to look into that.

I have a degree in political science and I am now taking a post graduation in public administration. I have had several courses on economics, but as far as math goes they have never gone beyond simple graphs. The occasional equation was not really mandotory for study.

For the time being, I am expecting to study on my own. Maziotis (talk) 09:31, 2 April 2011 (UTC)

Would I be confortable with the math in those articles after being able to study something like this?...[4] Just trying to see if i'm moving in the right direction.Maziotis (talk) 21:09, 2 April 2011 (UTC)

No that book won't be enough, you will need to know a little about differential equations, and more generally pick up a bit of sophistication (making sense of jargon in several fields) that's not exactly a matter of technical knowledge. But the stuff in those pdfs is not that advanced, it's just messy because there are a lot of (simple) equations, just like adding up 10-digit numbers is not really more advanced than adding 3-digit numbers. Maybe you don't want a math book per se, but something like "mathematical method for social sciences". WP:RDS might be a better place than this to ask for suggestions. 75.57.242.120 (talk) 01:53, 3 April 2011 (UTC)

Why not forward you question to the authors of the articles? They may be helpful. Bo Jacoby (talk) 07:44, 3 April 2011 (UTC).

I guess I will come to all that later. For now, I have to start with the basics. Does making sense of jargon in several fields means that I need to get involved in other mathematical fields than precalculus, elementary algebra, calculus and differential equations? Maziotis (talk) 11:08, 3 April 2011 (UTC)