I have been given two expressions:
$$\chi(\xi, P) = - Re \left[\sqrt[3]{\sqrt{\displaystyle P^3 - 3 \cdot \xi^2 \cdot P^2 + 3 \cdot \xi^4 \cdot P} + \xi^3-\xi} \cdot (1+ j \cdot \sqrt{3}) \right].$$ and $$W (\xi, m, \nu) = \frac{\nu}{2 \cdot \left[1 + \nu^2 \left(\xi - m \right)^2 \right]^{3/2}}.$$
You need to find:
$$\overline{\chi(P, m, \nu)} = \int\limits_{\xi_{min}}^{\xi_{max}} \chi(\xi, P) \cdot W(\xi, m, \nu) \cdot d\xi,$$
But since there are operations with complex numbers and roots in the first expression, I cannot calculate the integral. Since I know the ranges of variable changes: $P = -300 \dots -5 \; $, $\xi = 1 \dots 3$ I can take the value of $P = -150$ and plot the function of the first expression ($\chi(\xi, P = -150)$).
Now I'm trying to simplify the expression $\chi(\xi, P) $so that it can be used when calculating the integral.
I am trying to remove the terms from the expression that have little effect on the graph change. It is necessary to get rid of roots and complex numbers. But now I need help simplifying this expression, because in this form it is not possible to find a primitive one.
I 'm trying to add a summand under the cubic root:
$$ (1+j \sqrt{3})^3 = -8$$
then
$$\chi(\xi, P) = - Re \left[\sqrt[3]{\left(\sqrt{\displaystyle P^3 - 3 \cdot \xi^2 \cdot P^2 + 3 \cdot \xi^4 \cdot P} + \xi^3-\xi \right) \cdot (-8)} \right].$$
but already now I'm getting a discrepancy in the results:
