Can All Functions Be Integrated?

[deltabourne]

Is it actually impossible to integrate certain functions such as sin(x^2) or have we just not found a method yet?

 

[chroot]

Well, it's not possible to represent many integrals with elementary functions. Elementary functions, like log, sin, and arctan already have all their properties very well understood. The particular integral you mentioned,

\int \sin(x^2)\,dx

can be represented in terms of a Fresnel integral. The Fresnel integral is not an elementary function, and not all of its properties are understood yet. Every integral is representable using SOME yet-to-be-determined function, though. If you want to integrate some random function f(x), for example, you could simply say that

\int f(x)\,dx = F(x) + c

where F(x) is not yet known, or not yet well-understood. It almost feels like avoiding the question, but it's still a correct statement.

- Warren

 

[NSX]

Is it actually impossible to integrate certain functions such as sin(x^2) or have we just not found a method yet?

You could integrate it via. Taylor, and it gives you a pretty good approximation.

 

[HallsofIvy]

"sin(x²)" is integrable because it is continuous. Its anti-derivative is NOT an "elementary function" and no method, present or future, will make it one. (Although the definition of "elementary function" might change!)

 

[mathwonk]

this is a great question, and very interesting. To elaborate on Halls of Ivy's answer, if you read the fundamental theorem of calculus, it tells you that if f is ANY continuous function, then the area function for f, i.e. the indefinite integral of f taken from a fixed value of x like a, to x, is a function whose derivative is f. Thus any continuous f has an antiderivative, if you allow that an antiderivative can be a function as complicated as an area function. I.e. its value at x is given as a limit of Riemann sums, instead of by a nice simple formula only involving old familiar functions like sin and cos, or polynomials.

Thus since we know sin(x^2) is continuous, then the FTC says it has an antiderivative. But now comes the question of finding a nice formula for that antiderivative function. And that is not possible, because this function does not happen to be that closely related to any of our basic functions.

Nonetheless there are other things we can do to study this function even without a simple formula for it. We can approximate it, using Simpson's rule to approximate the area, or we can write a taylor series for it, stop at a certain point in the series, and
then just antidifferentiate that approximating polynomial, as suggested above.

The whole confusion here is in the meaning of the word "function". I.e. does the word mean a nice simple formula involving only familiar trig, exponential, and polynomial expressions, or is it any rule that can be described in words at all?


The familiar functions are called "elementary functions", and the more general kind are just called functions. The difficulty is that my students tend to think all functions are elementary. So it is common for my students on a test to state the fundamental theorem of calc, saying that every continuous function has its area function as an antiderivative, but still believe that some continuous functions like sin(x^2) have no antiderivative.

I will admit that I am not expert here, as I do not actually know how to prove that sin(x^2) is one of the functions with no elementary antiderivative, but I have noticed that almost no moderately complicated composition of different functions ever has one (except powers of elementary functions seem ok, but even sec^3 is pretty challenging, or even sec the first time you see it. also sin(arccos(x)) has an antiderivative). The theory of which functions have elementary antiderivatives is called differential algebra and I have a book on it but have never read it.

 

[eJavier]

Is it actually impossible to integrate certain functions such as sin(x^2) or have we just not found a method yet?

If by "integrate" you mean finding its anti derivative as a combinations of the functions we already know, then nope.

 

[JohnDubYa]

Take a sheet of paper, draw an x-y grid, and then draw an arbitrary curve. The curve obviously exists because you can see it, but what chance is there that the curve can be modeled exactly by a mathematical function?

Now consider a function f(x). If this function is continuous, then there exists a curve that models the area under the original curve for all values of x. (It is eay to see why this "area-producing" curve must exist.) But again, what chance is there that this area-producing curve can be modeled exactly by a mathematical function?

So how can we model the anti-derivative curve? Well, we can add an infinite number of curves such that the sum models the anti-derivative curve. One way to do this is to make the "sub-curves" polynomial terms, which means we are applying a polynomial expansion (such as a Taylor series).

 

[Dr.Brain]

ok actually this is not that tough to explain...a curve or a function is integrable for all real numbers if and onhly if the curve is differential and continuous all over the real line...if at any point say "a" the curve is not differential ...that is the curve has a peak or a corner at that point in the graph...then due to non-differential nature of the curve at "a",you cannot integrate because integration is basically reverse of differentiation...although you can do definite integration for selctive points but not indefinite integration

 

[Ethereal]

So do we say that a function which is the antiderivative of sin x^2 exists but we can't express it in terms of elementary functions we currently know? The question here is: Is it actually possible to express it in terms of elementary functions but that we don't know how to, or has it been proven it can't be expressed as such?

 

[shmoe]

Ethereal, it's been proven that it can't be expressed in terms of elementary functions, that is, no one will ever be able to it unless they use a different definition of "elementary function".


Dr.Brain, as I understand what you've written, it's not correct. A function does not have to be differentiable for it's integral to exist. Continuous is enough (more than enough actually). You can integrate functions with peaks. Try integrating |x|.

 

[mathwonk]

there are many different theories of integration, but for riemann integration, a function is riemann integrable on a finite interval [a,b] if and only if it is bounded and continuous except on a set of "measure" zero, which means that given any little interval no matter how short, it can be (in theory) cut into an infinite number of disjoint pieces that together will completely cover up the set of discontinuities. In fact there exist integrable functions that are not differentiable anywhere. look in spivak's calculus book, or apostol.

 

[gaurav89]

Try mathematica

To integrate sin(x^2) - try Mathematica - It gives the following -


Assume P = Sqrt(PI/2)
Then integral of sin(x^2) in the usual sense of a sum of a series comes out to P*FresnelS[x/P]

Now if you don't like the Fresnel functions - then one could say sin(x^2) can not be integrated. But then - someone may not like the Sine function itself...

Keeping formal integrability aside - the question about whether sin(x^2) can be integrated or not is really not a valid question in a true sense because it can be expanded in a series it can be integrated and its up to someone to give a name to it (Like Fresnel functions are in the above example) - and perhaps a more natural meaning to the resulting expression. Imagine that Log could not be defined in terms of exponents and was a fairly obscure function (Like Fresnel which only Optics people seem to care about ) ...Someone would have said 1/x can not be integrated..

 

[poolwin2001]

What are Fresnel integrals?

 

[TenaliRaman]

http://mathworld.wolfram.com/FresnelIntegrals.html

-- AI

 

[Feynman]

Use the numerical approximation

 

[HallsofIvy]

Now if you don't like the Fresnel functions - then one could say sin(x^2) can not be integrated. But then - someone may not like the Sine function itself...

True, but whether you like them or not, the "elementary functions" everyone has been talking about are pretty well (if arbitrarily) defined. Sine is an "elementery" function, the Fresnel functions (and the Bessel functions, the Erf function, Lambert's W function, etc.) are not!

Take a sheet of paper, draw an x-y grid, and then draw an arbitrary curve. The curve obviously exists because you can see it, but what chance is there that the curve can be modeled exactly by a mathematical function?

As long as no two points have the same y value, 100% because that the definition of "mathematical function". Again, it may not be a function anyone has ever defined before!