i need some inspiration for this integral. The question: Calculate for an n element N this improper R-Integral
$$ \int \frac {1} {\prod_{i=1}^n (1+x_{i}^2)} d(x_1,...x_n) $$
I am sitting here for 4 hours and i have no idea what to do. Thanks for your help!
Assuming you want $$ \int_0^\infty \frac {1} {\prod_{i=1}^n (1+x_{i}^2)} d(x_1,...x_n), $$ because the function is a product function, this is simply the product of the iterated interals: namely, $$ \int_A\int_Bf(x)g(y)\,dy\,dx=\left(\int_Af(x)\,dx\right)\left(\int_Bg(y)\,dy\right). $$ So $$ \int_0^\infty \frac {1} {\prod_{i=1}^n (1+x_{i}^2)} d(x_1,...x_n)=\prod_{j=1}^n\int_0^\infty\frac1{1+x^2}\,dx =\prod_{j=1}^n \frac\pi2=\left(\frac\pi2\right)^n. $$