The discussion centers on Liouville's extension of Dirichlet's theorem, exploring its applications, particularly in the context of integration. Participants inquire about the theorem's relevance and its potential use in evaluating specific integrals.
Participants express differing views on the applicability of Liouville's extension of Dirichlet's theorem to the integral in question, with some suggesting it is not relevant while others seek clarification on its use.
The discussion reveals uncertainty about the connection between Liouville's extension of Dirichlet's theorem and the specific integral posed, as well as the limitations of the theorem's applicability to certain types of integrals.
MarkFL said:I can help you evaluate that definite integral using elementary techniques if you would like.
Even so, I have moved this topic to Analysis instead. :D
You can find a statement of those theorems here. They provide a way to calculate triple integrals of certain functions over the region of three-dimensional space given by $x\geqslant0,\: y\geqslant0,\: z\geqslant0,\: x+y+z\leqslant1.$ I cannot see any way in which these results could have anything to do with an integral such as $$\int_0^{\pi/2} \!\!\!\!\!\!\cos^2(x)\sin^2(x)\,dx$$, which as MarkFL points out can be evaluated using far more elementary techniques.ksananthu said:What is liouville's extension of dirichlet's theorem ?
and where can I use such a theorem ?
Can I apply Integration like this ?
$$\int_0^{\frac{\pi}{2}} \cos^2(x)\sin^2(x)\,dx$$