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
- 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 ]
- 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 ]
- 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.
