MHB Liouville's extension of Dirichlet's theorem

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What is liouville's extension of dirichlet's theorem ?
and where can I use such a theorem ?
Can I apply Integration like this ?
$$\int_0^{\frac{\pi}{2}} \cos^2(x)\sin^2(x)\,dx$$
 
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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
 
Re: liouville's extension of dirichlet's theorem

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

Thank you.
But i want to know how we apply above theorem for integration like this
 
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 ?
$$\int_0^{\frac{\pi}{2}} \cos^2(x)\sin^2(x)\,dx$$
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.
 
I posted this question on math-stackexchange but apparently I asked something stupid and I was downvoted. I still don't have an answer to my question so I hope someone in here can help me or at least explain me why I am asking something stupid. I started studying Complex Analysis and came upon the following theorem which is a direct consequence of the Cauchy-Goursat theorem: Let ##f:D\to\mathbb{C}## be an anlytic function over a simply connected region ##D##. If ##a## and ##z## are part of...
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