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## \Delta U = - \int \vec F \cdot d \vec r ##. Since the gravitational force is in the same direction as the path of the particle, ##\Delta U = U(b) = - \int \vec F \cdot d \vec r = - \int F dr = - \int_\infty^b \frac{GMm}{r^2} dr = -GMm(-\frac{1}{b}) = \frac{GMm}{b} ##. Where is my mistake?