What is the integral of sec^2(x) dx?

The integral of sec²(x) dx is a fundamental result in calculus linked to the derivative of the tangent function. Specifically, the derivative of tan(x) with respect to x is sec²(x). Therefore, when finding the integral of sec²(x), we are essentially looking for the function whose derivative gives us sec²(x).

Since tan(x) is the function that fulfills this condition, the integral of sec²(x) dx equates to tan(x) plus a constant of integration, represented as C. This reflects the general nature of indefinite integrals, where we include an arbitrary constant to account for all possible antiderivatives.

In contrast to the other functions listed, such as sec(x), cot(x), and csc(x), their derivatives do not yield sec²(x), making them incorrect choices in this context. Thus, the identification of tan(x) with its corresponding integral is vital in calculus and serves to reinforce the connection between integration and differentiation.