First of all, from a mathematical point of view, Euler angle integral is an important method to solve the problem of rigid body motion. Rigid body movement refers to the movement in which the relative positions of points in an object remain unchanged during the movement. Euler angles are three angles that describe the motion state of rigid bodies, and are usually expressed by Euler angles (rotation angles around X-axis, Y-axis and Z-axis). Through Euler angle integration, the position and attitude of a rigid body at any time can be solved. This is of great significance to spacecraft, robots and other engineering applications.

Secondly, from the physical point of view, Euler angle integral is widely used to describe the motion of objects in three-dimensional space. For example, in astronomy, Euler angle integral is needed to study the trajectory of planets, stars and other celestial bodies; In the aerospace field, Euler angle integral is also needed to study the attitude control and navigation of aircraft. In addition, Euler angle integral also plays an important role in computer graphics, virtual reality and other fields.

But there are some problems in Euler angle integral. Because of the discontinuity of Euler angle, when an object changes from one attitude to another, the so-called "universal lock" phenomenon will appear, which will lead to the divergence of integral. In order to solve this problem, researchers have proposed many methods, such as quaternion method and directional cosine method. These methods overcome the shortcomings of Euler angle integral to some extent, and make Euler angle integral more reliable in practical application.