How Do We Solve the Definite Integral of log(sin(x))*log(cos(x)) from 0 to pi/2?

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SUMMARY

The definite integral of log(sin(x)) * log(cos(x)) from 0 to π/2 is a complex problem that has been addressed in mathematical literature, notably by Ramanujan. The integral is not elementary and can be expressed in terms of the zeta function. A suggested approach involves using infinite products for sine and cosine, along with the logarithmic identity log(ab) = log(a) + log(b). Reference to "Treatise on Integral Calculus Vol. 2" by Joseph Edwards highlights a solution by Wolstenholme.

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How should we proceed to find the definite integral


Int[ log(sin(x))*log(cos(x)) ,{x,0,pi/2} ] ?






mathslover
 
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I can't remember the source of even the exact answer :( But I've seen integrals of that form done in some papers by Ramanujan where he expressed the answer in terms of the zeta function and something else, so it's not exactly an elementary integral.

This isn't how he did it, and I'm not sure if it will help, but you could express cos/sin in terms of an infinite product, split up the logs ( log [ab]= log [a] + log ) and then continue?
 
Leafing through "Treatise on Integral Calculus Vol. 2 --Joseph Edwards (1922)",I found
Wolstenholme had solved the above problem nicely as follow:

-Ng
 

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