Integrate (ln ln x)^n

1. Apr 16, 2009

flouran

How do you integrate (ln ln x)^n for any n?

2. Apr 17, 2009

nicksauce

Well since mathematica isn't able to find a formula for n=2, I'm going to say "numerically".

3. Apr 17, 2009

flouran

What do you mean exactly by "numerically"? Do you mean that i should evaluate it as a definite integral?

4. Apr 17, 2009

nicksauce

Well I just mean if you want to find a general closed for expression for the indefinite integral, you are out of luck. Therefore the only way I can conceive of doing an integral with this expression would be to do a definite integral numerically.

5. Apr 17, 2009

22/7

there is one place where the integral can be evaluated explicitly that I know of.
$$\int$$-log[-log[x]]dx=Euler Constant (.577...)
This follows from differentionation the gamma function in its product and integral forms and making a change of variables.

6. Apr 17, 2009

flouran

Although this method would be painful, since I cannot express this integral in terms of elementary functions, could I represent (ln ln x)^(n) as a Taylor polynomial (what is the Taylor series for ln ln x, anyways?) and then integrate that and leave it as a Taylor Series?