# What is the antiderivative of sec(x)?

#### Understand the Problem

The question is asking for the antiderivative (or integral) of the secant function, which involves finding a function whose derivative is sec(x). This requires knowledge of integral calculus.

\ln| \sec(x) + \tan(x) | + C

The antiderivative of $\sec(x)$ is $\ln| \sec(x) + \tan(x) | + C$

#### Steps to Solve

1. Use an integral identity for secant

To find the antiderivative of $\sec(x)$, we use a known integral identity. [ \int \sec(x) , dx = \ln| \sec(x) + \tan(x) | + C ]

1. Substitute and simplify

Substitute $\sec(x)$ and $\tan(x)$ into the integral formula. The antiderivative (or integral) of $\sec(x)$ is: [ \ln| \sec(x) + \tan(x) | + C ]

1. Include the constant of integration

Don't forget to add the constant of integration, $C$, as it represents the family of all antiderivatives.

So the final answer is: [ \ln| \sec(x) + \tan(x) | + C ]

The antiderivative of $\sec(x)$ is $\ln| \sec(x) + \tan(x) | + C$

A common mistake is to forget the absolute value when writing the logarithmic function. Always include $| ... |$ to ensure correctness.