How to find the original function of cot?To require the original function of cot(x), we need to find a function f(x) that satisfies f'（x）=cot（x）。This problem can be solved by some common methods, such as through spins and integrals.
A common approach is to rotate cot(x) and convert it into the form of other known functions. We know that cot(x) can be represented as cos(x) sin(x). By transforming cot(x)=cos(x) sin(x), we can get an equivalent form of integration, which is (cos(x) sin(x))dx. Now we convert sin(x) to u, then do a geometric rotation of u, convert cos(x) to -du, and do variable substitution. In this way, we get a new integral expression (-du u). Solving for this new integral expression, we get -ln(|u|+c, where c is a constant. Finally, reducing u to sin(x), we get the original function of cot(x) as -ln(|sin（x）|）c。
Another way is to solve using the inverse of the derivative-defined operation, the integral. We observe that the derivatives of cot(x) and cot(-x) are the same, so we can express the primitive function of cot(-x) as -c (constant) plus cot(x). Next, we need to solve the problem of the primitive function of cot(0). Within the domain of cot(0), its original function is discontinuous. We can divide the primitive function of cot(x) into different parts according to the different domains in which it is defined: the part between 0 and , its primitive function is -ln(|sin（x）|）c；In the part between 2 and , its original function is -ln(|sin（x）|）d。In this way, we can find the original function of cot(x).
To sum up, the primitive function of cot(x) can be solved by a variety of methods, including rotation and integration. We can either convert cot(x) into the form of other known functions, or solve it using the inverse of the derivative-defined operation. Regardless of which approach is taken, we need to pay attention to the discontinuities of the defined domains and solve the original functions of cot(x) in different defined domains separately.