Integral from 0 to Infinity of 1/(sqrt(x)(1+x^2)) dx

[@maxvangulik1988]

x=tan(Ø)
dx=sec^2(Ø)dØ
I=int[0,pi/2](sqrt(cot(Ø)))dØ
int[0,pi/2](sin^(2u-1)(Ø)cos^(2v-1)(Ø))dØ=1/2•ẞ(u,v)
I=1/2•ẞ(1/4,3/4)
ẞ(x,1-x)=pi•csc(pi•x)
I=pi/2•csc(pi/4)
I=pi/sqrt(2)