Discussion Overview
The discussion revolves around evaluating the integral
$\displaystyle \int{\sin{x}\cos{\left(\cos{x}\right)}}dx$. Participants explore various substitution methods and share their thoughts on the integration process, focusing on the anti-derivative of the given function.
Discussion Character
- Homework-related
- Mathematical reasoning
Main Points Raised
- One participant expresses confusion about how to approach the integral, suggesting a possible relationship between $\cos{\left(\cos{x}\right)}$ and the derivative of $\sin{x}$.
- Several participants propose the substitution $u=\cos(x)$, with one noting that this leads to a more straightforward integral.
- Another participant reiterates the substitution and provides the differential $du=-\sin(x)dx$ as part of their response.
- There is a correction regarding a sign error in the integration result, with a participant emphasizing the importance of including the constant of integration and back-substituting for $u$.
- Participants confirm the integration result as $-\sin(u) + C$, with a focus on the need to revert to the original variable $x$.
Areas of Agreement / Disagreement
Participants generally agree on the substitution method and the resulting integral, but there is a noted correction regarding a sign error. The discussion remains focused on the integration process without reaching a final consensus on the complete solution.
Contextual Notes
Some participants mention the need to back-substitute for $u$, indicating that the discussion is still in the process of finding the anti-derivative in terms of the original variable.
