Can the Integration of x^x be Solved Conventionally?

[Quadratic]

A few of my friends and I have been trying to integrate/derive the following:

f(x) = x^x

without success. I'm not sure if it can be done conventionally, but I was wondering if anyone had any thoughts on this one. Thanks.

 

[TD]

It cannot be done "conventionally", i.e. you cannot express its primitive in a closed form of elementary functions, which is also the case for e^(x²), sqrt(sin(x)), sin(x)/x, ...

 

[benorin]

While $$\int x^x dx$$ cannot be expressed in a finite number of elementary functions, derivative of $$f(x)= x^x$$ can be obtained by logarithmic differentiation: take the log of both sides to get

$$ln[f(x)]=ln\left( x^x\right) = x ln(x)$$

now differentiate both sides to get

$$\frac{f^{\prime}(x)}{f(x)}= ln(x)+1$$

multiply by f(x) to get

$$f^{\prime}(x)= f(x)( ln(x)+1) = x^x( ln(x)+1)$$

 

[gnomedt]

Attached is a graph of $$y=\int_0^x u^u du$$, courtesy of Apple Grapher.