Contour plot implicit equation with conditions?

I want to contour plot x versus T but don’t know how to plot it. Appreciate your help.

HW9 P3.nb (3.05 KB)

can you calculate x for a specific value of T?

Sure, I can calculate the x for every T, but how to correlate all this equation and plot it?

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int = Integrate[1/(x^3 - 1), x];
Map[Framed, int, Infinity]