Ramanujan Log-Trigonometric Integrals

Oloa (2010, pers. comm., Jan. 20, 2010) has considered the following integrals containing nested radicals of 1/2 plus terms in  and :

			(1)
			(2)

which he terms Ramanujan log-trigonometric integrals because they involve terms like Ramanujan's nested radicals of 1/2.

The special case

(3)

was known to Euler.

Amazingly, the general integrals have closed forms

			(4)
			(5)

for .

REFERENCES:

Oloa, O. "Two Conjectures on Ramanujan Log-trigonometric Integrals." Unpublished manuscript. January 2010.