Let the complex variable function f (z) = 1/(z 4+ 1), obviously the function I * * * has four first-order poles {a(k)}.

a(k)=e^[(&； π+2k & amp； pi)i/4]

Then, using the P(z)/Q(z) residue calculation theorem.

resf(z)= 1/4z^3=-a(k)/4,z=a(k),k=0, 1,2,3

Because f(z) has only two poles a(0) and a( 1) in the upper half plane, so

∫ 1/( 1+x^4)dx x∈(0，+∞)

=-∏i{e^[(∏i/4)+e^(3∏i/4)]}

=∏sin(∏/4)/2

=∏/2√2