i want the answer of integration of a function please help?

Question

"Integration of X raised to the power of x with respect to X ?"
all my teachers R unable to answer this question please e-mail me
the solution as soon as possible.....

Answer 1

There is no closed form for this integral. It is one of a fairly large family of functions whose integrals cannot be expressed by a finite combination of the usual 'elementary' functions like the trig functions, exponentials, logarithms, etc. We know it is actually impossible to do this through something called 'differential Galois theory', which characterizes those functions that do have elementary anti-derivatives,

Other functions which don't have nice anti-derivatives:
[sin x]/x
exp(-x^2)
[e^x]/x
This list goes on...

Answer 2

I don't think you will get an answer in closed form. Variable exponents tend to be difficult anyway, and in your case the base varies as well...

We can explore a little bit though. The derivative of an exponential function always contains that exponential function, and the same is the case here.

[1] ... f(x) = x^x = exp(x * ln(x)) ==>
[2] ... f'(x) = exp(x * ln(x)) * [ln (x) + 1]

so the derivative is f'(x) = f(x) * [1 + ln(x)]

This suggests that the integral of f(x) is equal to

[3] ... INT f(x) dx = f(x) - INT ln(x)*f(x) dx

This is not really a solution... but I hope I have entertained you and convinced you that it is not so easy.

Answer 3

Dutch_ is correct. This is a nasty integral with no formula solution. However Scott R (on this site) gave a good answer to this question which was raised by someone else about a week ago.

Here's the link to Scott's answer:

http://answers.yahoo.com/question/?qid=20060718134426AAnUACN

Here's another link that provides and entertaining thread discussion of this problem:

http://sciforums.com/showthread.php?t=37958

Answer 4

I think there is no direct way to integrate this one.. but you can aproximate the value.

Answer 5

Best thing to do here is use numeric techniques.

Answer 6

x^x = e^(xln(x))
then integrate by parts

Answer 7

I don't think that there is one