Liouville's extension of Dirichlet's theorem provides a framework for evaluating certain types of integrals, particularly in the context of triple integrals over specified regions in three-dimensional space. The theorem is applicable in scenarios where functions are integrated over domains defined by inequalities, such as $x \geq 0$, $y \geq 0$, $z \geq 0$, and $x + y + z \leq 1$. However, for specific integrals like $$\int_0^{\frac{\pi}{2}} \cos^2(x)\sin^2(x)\,dx$$, simpler elementary techniques are recommended for evaluation, as they yield results without the need for advanced theorems.
PREREQUISITESMathematicians, calculus students, and educators seeking to deepen their understanding of advanced integration techniques and the applications of Liouville's extension of Dirichlet's theorem in mathematical analysis.
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 ?
$$\int_0^{\frac{\pi}{2}} \cos^2(x)\sin^2(x)\,dx$$