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Liouville's extension of Dirichlet's theorem

ksananthu

New member
Jul 14, 2013
5
What is liouville's extension of dirichlet's theorem ?
and where can I use such a theorem ?
Can I apply Integration like this ?
\(\displaystyle \int_0^{\frac{\pi}{2}} \cos^2(x)\sin^2(x)\,dx\)
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Re: liouville's extension of dirichlet's theorem

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
 

ksananthu

New member
Jul 14, 2013
5
Re: liouville's extension of dirichlet's theorem

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
Thank you.
But i want to know how we apply above theorem for integration like this
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,701
What is liouville's extension of dirichlet's theorem ?
and where can I use such a theorem ?
Can I apply Integration like this ?
\(\displaystyle \int_0^{\frac{\pi}{2}} \cos^2(x)\sin^2(x)\,dx\)
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 \(\displaystyle \int_0^{\pi/2} \!\!\!\!\!\!\cos^2(x)\sin^2(x)\,dx\), which as MarkFL points out can be evaluated using far more elementary techniques.