Liouville's extension of Dirichlet's theorem

[ksananthu]

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]

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]

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]

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.

 

[blue_raver22]

Liouville's extension of Dirichlet's theorem is a mathematical theorem that states that any algebraic number is either a rational number or a transcendental number. This is an extension of Dirichlet's theorem, which only applies to irrational numbers. In other words, Liouville's theorem provides a way to determine if a given number is algebraic or transcendental.

This theorem has many applications in number theory and algebraic geometry. It can be used to prove the transcendence of certain numbers, such as e and pi, and to study the properties of algebraic numbers.

As for your question about applying integration using Liouville's theorem, it is not directly applicable. Liouville's theorem deals with the nature of numbers and not with their integration. However, it can be used in conjunction with other mathematical techniques to solve integration problems involving transcendental functions.