Integrate by parts:
#int lnx/x^(1/2) dx = int lnx * x^(-1/2)dx#
#int lnx/x^(1/2) dx = 2 int lnx * d/dx (x^(1/2)) dx#
#int lnx/x^(1/2) dx = 2 x^(1/2)lnx - 2 int x^(1/2)*d/dx(lnx)dx#
#int lnx/x^(1/2) dx = 2 x^(1/2)lnx - 2 int x^(1/2)*1/xdx#
#int lnx/x^(1/2) dx = 2 x^(1/2)lnx - 2 int x^(-1/2)dx#
#int lnx/x^(1/2) dx = 2 x^(1/2)lnx - 4 x^(+1/2)+C#
#int lnx/x^(1/2) dx = 2 x^(1/2)(lnx - 2)+C#