Integration involving the complete elliptic integral of the first kind K(k)? Is there any reference showing how to do definite integrals involving the complete elliptic integral of the first kind K(k)? 
Something like 


*

*$\int_0^1 K(k) dk $

*$\int_0^1 k^nK(k) dk$ 

*$\int_0^1 \frac{K(k)}{1+k} dk $


etc...Thanks a lot.
 A: Question bumped to the home page by Community ...
No answer yet, I guess.  (Dan has not been here for over 6 years, so I doubt he will ever choose an answer.)
Gradshteyn & Ryzhik
6.141.1:
$$
\int_0^1 \mathbf{K}(k)\;dk = 2G
$$
where $G$ is Catalan's constant.  
6.144:
$$
\int_0^1 \frac{\mathbf{K}(k)}{1+k}\;dk = \frac{\pi^2}{8}
$$
6.146 is a reduction formula, reducing the exponent by $2$:
$$
n^2 \int_0^1 k^n \mathbf{K}(k)\;dk
 = (n-1)^2\int_0^1 k^{n-2} \mathbf{K}(k)\;dk + 1
$$  
I don't see the odd powers of $k$ directly.  Maybe you can get them using   
6.142.1:
$$
\int_0^1 \left(\mathbf{K}(k) - \frac{\pi}{2}\right)\;\frac{dk}{k} = \pi\log 2 - 2 G,
$$
6.142.2:
$$
\int_0^1 \left(\mathbf{K}(k) - \frac{\pi}{2}\right)\;\frac{dk}{k^2} = \frac{\pi}{2} - 1
$$
A: A. Dieckmann compiled a big table of indefinite and definite integrals envolving non-elementary functions. See the following links: 
http://www-elsa.physik.uni-bonn.de/~dieckman/IntegralsIndefinite/IndefInt.html
and 
http://www-elsa.physik.uni-bonn.de/~dieckman/IntegralsDefinite/DefInt.html 
