What is the integral of sec(x)tan(x) dx?

To find the integral of sec(x)tan(x) dx, it is helpful to recognize the relationship between the derivatives of trigonometric functions and how they can guide us to the correct integral.

The derivative of sec(x) is sec(x)tan(x). This means that when we differentiate sec(x), we obtain sec(x)tan(x) as a result. Therefore, when computing the integral of sec(x)tan(x), we inherently find that its antiderivative is sec(x).

Thus, the integral of sec(x)tan(x) dx is indeed sec(x) + C, where C represents the constant of integration. Understanding this relationship—knowing that the integral is directly related to the derivative of sec(x)—is crucial in determining the correct answer.

This connection also allows for a clear identification of what other choices represent. For instance, while the integral of tan(x) results in -ln|cos(x)| + C, and the integral of sec^2(x) dx equals tan(x) + C, these do not relate directly to sec(x)tan(x). Additionally, the expression ln|sec(x) + tan(x)| + C is connected to the integral of sec(x), but it arises from a different