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.

Answer

\ln| \sec(x) + \tan(x) | + C
Answer for screen readers

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 $

More Information

The integral of secant function results in a logarithmic function, which might be unexpected for some learners.

Tips

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

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