# 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

1. $\int_0^1 K(k) dk$
2. $\int_0^1 k^nK(k) dk$
3. $\int_0^1 \frac{K(k)}{1+k} dk$

etc...Thanks a lot.

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This is Chapter 6.1 of Gradshteyn, I.S. & Ryzhik, I.M. (2007) "Table of Integrals, Series and Products", 7th edn., Amsterdam: Academic Press. – Aleksey Pichugin Aug 30 '10 at 21:24
Anyway... it's good you (seem to have) specified that you were using the modulus $k$ instead of the parameter $m=k^2$ as the argument for your elliptic integrals. – J. M. Aug 30 '10 at 22:24
The first and second one can be handled by the techniques in this paper: repository.cmu.edu/cgi/… ; the last one does not seem to have a nice closed form. – J. M. Aug 31 '10 at 7:42
J. Mangaldan, all of these integrals are available in Gradshteyn & Ryzhik. In particular, the last one is given by eq. (6.144): $$\int_0^1\frac{K(k)}{1+k}dk=\frac{\pi^2}{8}.$$ – Aleksey Pichugin Aug 31 '10 at 8:43
Hmm, too bad I didn't have my copy of G&R nearby to check the third one, but the first and second one are easily handled as integrals of hypergeometric functions. Thanks for the pointer Aleksey! – J. M. Aug 31 '10 at 12:10