# Liouville's extension of Dirichlet's theorem

#### ksananthu

##### New member
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

Staff member
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. #### ksananthu

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

#### Opalg

##### MHB Oldtimer
Staff member
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.