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while I'm reading this paper. I came to the question, if the integral of the angular velocity in the inertia frame and the integral of the euler angles are the same?

Equation (35) describes a rotation from inertia frame to the body frame:

$$\mathbb{z}' = R_{ijk}(\mathbb{u}) \mathbb{z}$$ So that means, that the euler angles are the angles between the inertia frame and the body frame.

Equation (39) describes the relation between the euler angular rate and the inertia angular rate: $$\omega = E_{ijk}(\mathbb{u}) \dot{\mathbb{u}}$$

So why integrating $\omega$ is not the same than the euler angles?

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