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

[mahesh_2961]

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

 

[DeadWolfe]

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

There are many such integrals of this nature.

 

[mahesh_2961]

Thanx wolfe, can u tell me where i can find such functions

Mahesh

 

[dextercioby]

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 $\sqrt{P(x)}$,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".

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.