Why Can't Some Integrals Be Evaluated Exactly?

[pivoxa15]

Homework Statement

Why are some integrals not able to be evaluated?

i.e. the integral of (1+1/x^4)^(1/2) is impossible to evaluate exactly from 0 to 1. Why??

 

[Hurkyl]

It's hard to answer your question without knowing just what you mean by "evaluated", "able", and "why".

 

[HallsofIvy]

If you mean "find an expression for the anti-derivative in an elementary form", that is true for almost all integrable functions. the problem is that we simply don't know enough functions. By "elementary functions" we typically mean rational functions, radicals, exponentials and logarithms, and trig functions. That is just a tiny part of all possible functions, even all possible analytic functions.

In a deeper sense, the problem is that while we have "formula" for the derivative, there is no "formula" for the anti-derivative; it is simply defined as the "inverse" of the derivative. And "inverses" are typically very difficult. If we define y= x^5- 3x^3+ 4x^3- 5x^2+ x- 7, the direct problem, to "evaluate" the function (Given x, what is y?) is relatively simple. The "inverse" problem, to "solve the equation" (Givey y, what is x) is much harder

 

[pivoxa15]

It's hard to answer your question without knowing just what you mean by "evaluated", "able", and "why".

evaluate by integrating from 0 to 1. which I have added in the OP.

able as you computing it exactly.

why as in why is it not able to be computed exactly.

 

[pivoxa15]

If you mean "find an expression for the anti-derivative in an elementary form", that is true for almost all integrable functions. the problem is that we simply don't know enough functions. By "elementary functions" we typically mean rational functions, radicals, exponentials and logarithms, and trig functions. That is just a tiny part of all possible functions, even all possible analytic functions.

In a deeper sense, the problem is that while we have "formula" for the derivative, there is no "formula" for the anti-derivative; it is simply defined as the "inverse" of the derivative. And "inverses" are typically very difficult. If we define y= x^5- 3x^3+ 4x^3- 5x^2+ x- 7, the direct problem, to "evaluate" the function (Given x, what is y?) is relatively simple. The "inverse" problem, to "solve the equation" (Givey y, what is x) is much harder

So we will have an infinite series as an antiderivative for the function? If so then we can compute the series? It might diverge?

 

[morphism]

Well, your example diverges.

\int_0^1 \sqrt{1 + \frac{1}{x^4}} \, dx = \int_0^1 \frac{\sqrt{x^4 + 1}}{x^2} \, dx \geq \int_0^1 \frac{1}{x^2} = \infty

 

[HallsofIvy]

So we will have an infinite series as an antiderivative for the function? If so then we can compute the series? It might diverge?

Given an analytic function to be integrated, we certainly can calculate its Taylor's series and integrate that term by term to get a power series for the anti-derivative. If I remember correctly, it should converge on the same radius of convergence as the original function.

Of course, not all functions of interest are analytic- i.e. have a Taylors series that converges TO the function on some interval around the central point.

 

[Hurkyl]

evaluate by integrating from 0 to 1. which I have added in the OP.

able as you computing it exactly.

why as in why is it not able to be computed exactly.

What precisely do you mean by "compute"?

If you mean "write in decimal notation with finitely many digits", then isn't the answer obvious?

If you mean "write down an algorithm that, given an integer n, outputs the n-th digit in its decimal representation", then it is computable.