What Is the Integral of Functions Like x^x, e^[x^2], and cos[x]^[sin[x]?

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The integral of functions like x^x, e^[x^2], and cos[x]^[sin[x] cannot be expressed in terms of elementary functions. The integral of e^[x^2] can be related to special functions, specifically the error function with an imaginary argument. Attempts to simplify x^x using integration by parts lead to complex expressions that are not solvable. Overall, these integrals present significant challenges and do not yield straightforward solutions. The discussion highlights the complexity of integrating such functions in calculus.
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i came acreoss a problem that's red like this ,wats the integral of x^x i gev it all i could so can anyone help me,furthermore wats the integral ,of a fuction raised 2 another e.g x^x,e^[x^2],cos[x]^[sin[x].and so on
 
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One of them,viz.\int \mbox{exp} \left(x^{2}\right) \ dx,can be expressed as function of the "common" special functions (the erf of imaginary argument).The other that u mentioned cannot...


Daniel.
 
Where are you getting these problems? None of these functions can be integrated in terms of elementary functions.
 
Just a quick thought: what about breaking x^x into a product. For instance x^x = [x^(x-1)]*x^1 and then try integration by parts. It's been a while since my calculus days but that sounds like something I might have tried.
 
It's usless.Here's why.

\int x^{x} \ dx =\int x^{x-1} x \ dx=\frac{x^{x-1}x^{2}}{2}-\frac{1}{2}\int x^{x-1}\left[x(x-1)+x^{2}\ln x\right] \ dx

and the second integral looks horrible...


Daniel.
 
hey thnxs for all but they still do not help ,
 
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