What is the Antiderivative of Sec?

The antiderivative of the secant function, denoted as sec(x), is a common question in calculus. The antiderivative of sec(x) is given by:

∫ sec(x) dx = ln|sec(x) + tan(x)| + C

Here, ln represents the natural logarithm, and C is the constant of integration. This result can be derived using integration techniques such as substitution and trigonometric identities.

To understand why this is the case, let’s break it down:

1. Rewrite sec(x): The secant function can be rewritten as sec(x) = 1/cos(x).
2. Multiply by a form of 1: Multiply the integrand by (sec(x) + tan(x))/(sec(x) + tan(x)), which is a form of 1.
3. Substitute: Let u = sec(x) + tan(x). Then, du = (sec(x)tan(x) + sec²(x)) dx.
4. Integrate: The integral simplifies to ∫ (1/u) du, which is ln|u| + C.
5. Substitute back: Replace u with sec(x) + tan(x) to get the final result.

This antiderivative is useful in various applications, including solving differential equations and evaluating definite integrals involving the secant function.