integration problem

I don't think you can actually integrate this, but you can probably say that f(x) = x^x, and F, which is after integrating, F(x) = int ( t^t, t, a, x)
where a is a constant, then when you differentiate F(x), you can apply the Fundamental Theorem of Calculus, then you get f(x). Hope it helps.

2015-07-04 15:57:06 補充：
5^x = e^ ln(5^x) = e^ (ln5)*x

then int (5^x) = int ( e^ (ln5)*x )
then ln5 is just a constant
so int ( e^ (ln5)*x) = e^(ln5)/(ln5)

this is a rule for integral of a^x, if you want a clearer sense of what is happening,
let u=(ln5)x, du=ln5 dx
then int(e ^ (ln5)x ) = 1/(ln5) int ( e^u du)
then you get (e^u)/(ln5), sub back ln5=u, you get the same thing

2015-07-04 19:36:38 補充：
Yea...forgot to do that step back, sorry! But you get the idea right?

Let y = 5^x
So, ln y = x ln 5
(1/y) dy = (ln 5) dx
==> dx＝[1/(y ln 5)] dy
Therefore,
∫ 5^x dx
＝∫ y [1/y ln 5)] dy
＝(1/ln 5) ∫ dy
＝y/ln 5＋C
＝5^x/ln 5＋C

btw,
∫ x^x dx + ∫ x^x ln x dx 會易D
∫ x^x dx + ∫ x^x ln x dx = x^x + C

答案很複雜，請看：
http://www.wolframalpha.com/input/?i=int%20x%5Ex%20dx&t=crmtb01

相信你只是為好奇而問這問題。

2015-07-04 19:41:53 補充：
For a > 0, ∫ a^x dx = a^x / ln(a) + C

2015-07-04 19:42:19 補充：
"and a ≠ 1" for the above.