Are there other non-integrable functions besides x^x?

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In summary, the function x^x does not have any indefinite integral expressible in elementary functions. One must use numerical methods to find a definite integral. There are many other functions of this nature, such as elliptic integrals, that do not have a "cute" antiderivative. Mathematicians refer to these types of functions as "nonelementary" and there is no clear definition for them. When an antiderivative cannot be found, other tools such as numerical analysis or expansion in Taylor series can be used. The books by Abramowitz & Stegun and Gradsteyn & Rytzhik can be helpful in these cases. If the antiderivatives exist but cannot be expressed in terms of "elementary
  • #1
mahesh_2961
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hai
i heard that the function x^x doesn't have any indefinite integral and hence one can't find definite integral by normal methods .. So one has to go for numerical methods , i haven't tried this ...
just curious to know if there exist more fuctions in the same class ...

regards
mahesh :smile:
 
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Well, the integral "exists" of course, but it's not the expressible in elementary functions.

There are many such integrals of this nature.
 
  • #3
Thanx wolfe, can u tell me where i can find such functions

Mahesh
 
  • #4
DeadWolfe said:
Well, the integral "exists" of course, but it's not the expressible in elementary functions.

There are many such integrals of this nature.

Yap,just about elliptic integrals.Basically most of the type [itex] \sqrt{P(x)} [/itex],where P(x) is a polynomial with real coeffcients of degree larger of equal with 3,get the "chance" of not having a "cute" antiderivative.Mathematicians invented the famous syntagma "nonelementary function",referring to this sort of functions which come up when searching for antiderivatives.They couldn't come up with a decent definition for this "nonelementary". :tongue2:

Anyway,when you spot something wrong,i.e.u can't find an antiderivative,try for other tools.Numerical analysis works,but only in the case on definite integral,where the result is a number.Sometimes,u can expand the integrand in Taylor series (though the ray may be small) or express it terms on tabulated "nonelementary functions".The books by Abramowitz & Stegun and Gradsteyn & Rytzhik may turn out to be handy.

Daniel.

PS.If the antiderivatives exist,but cannot be expressed in terms of "elementary" functions,then the function which makes up the integrand is integrable.
 
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1. What is a non-integrable function?

A non-integrable function is a mathematical function that cannot be integrated using traditional integration techniques, such as the definite or indefinite integral. These functions do not have a closed form solution and require more advanced mathematical techniques to be integrated.

2. How can I identify a non-integrable function?

One way to identify a non-integrable function is by looking for functions that contain special functions, such as trigonometric functions, logarithms, or exponential functions. Another way is to try to integrate the function using traditional techniques - if the integral cannot be evaluated, then the function is non-integrable.

3. Are there any benefits to studying non-integrable functions?

Yes, studying non-integrable functions can lead to a deeper understanding of mathematical concepts and techniques, as well as practical applications in fields such as physics, engineering, and economics. Non-integrable functions also play a crucial role in the development of more advanced mathematical methods.

4. Can non-integrable functions be approximated?

Yes, non-integrable functions can be approximated using numerical integration techniques, such as the trapezoidal rule or Simpson's rule. These methods use discrete data points to estimate the area under the curve of the function, providing an approximation of the integral.

5. Are there any real-life examples of non-integrable functions?

Yes, there are many real-life examples of non-integrable functions. For instance, the motion of a pendulum cannot be described by a closed-form function and requires advanced mathematical techniques to be integrated. Other examples include the Black-Scholes equation used in finance and the Schrödinger equation used in quantum mechanics.

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