Evaluate the integral of $ sec(x) $ along the unit circle
Answer 1
Matthew Carter
To evaluate the integral of $ \sec(x) $ along the unit circle, we consider the parametrization of the unit circle. The unit circle can be parametrized as $ x = \cos(\theta) $ and $ y = \sin(\theta) $, where $ \theta $ ranges from $ 0 $ to $ 2\pi $.
The integral to evaluate becomes:
$ \int_0^{2\pi} \sec(\cos(\theta)) \frac{d\theta}{d \theta} \ d\theta $
We need to express $ \sec(\cos(\theta)) $ in terms of $ \theta $. However, since $ \sec(x) $ is not straightforward to integrate on the unit circle, it is more practical to use a different approach, often involving complex analysis or residue theorem.
Answer 2
Ella Lewis
To evaluate the integral of $ sec(x) $ along the unit circle, consider the parametrization $ x = cos( heta) $ and $ y = sin( heta) $, with $ heta $ from $ 0 $ to $ 2pi $.
The integral transforms to:
$ int_0^{2pi} sec(cos( heta)) d heta $
This integral is complex and requires advanced techniques such as the residue theorem in complex analysis for evaluation.
Answer 3
Matthew Carter
To evaluate the integral of $ sec(x) $ along the unit circle, we use:
$ int_0^{2pi} sec(cos( heta)) d heta $
This integral is evaluated using complex analysis techniques.
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