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 Problem Statement

The question includes programming, but the basic problem is related to mathematics, that's why I posted it here.
In fact, I'm working on a condensed matter physics paper,where I need to draw integral
##g(\nu,2k_fz)## in units of ##(2k_f)^{1}## and interms of z.
##g(\nu,2k_fz)=\int_{1}^{1}\dfrac{\nu^2(1u^2)usin(2k_fzu)}{u^2\nu^2/2+u\sqrt{u^2\nu^2}}\, du##, where ##\nu=0.5##
I'm using Simpson's method to evaluate this integral. The problem is that the sign inside the the radical in the denominator will be negative while evaluating this method, that's why we need to treat the integrand as a complex function. The program(python) will plot for me just the real part of the result.
But my main question is: Is it Ok to use Simpson's method in order to evaluate integrals that their integrand is a complex function?
because the graph I'm getting is not even close to the graph depicted in the paper.
Is contour integration a better choice?
 Relevant Equations
 Simpson's method :