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

[abia ubong]

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

 

[dextercioby]

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.

 

[HallsofIvy]

Where are you getting these problems? None of these functions can be integrated in terms of elementary functions.

 

[dfollett76]

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.

 

[dextercioby]

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.

 

[abia ubong]

hey thnxs for all but they still do not help ,