Question 4.3 asks for the heat capactiy in the low temperature limit. The solution for this exercise describes taking the inegral up kappa_{D}. This seems like the solution for medium T, why cant we use the upper boundary of infinity here?
You’re right, the solution is a bit more general than the question, and compares the full expression. Integrating to infinity would be enough.
Is that also the case in the following question from the exam?
Here i also expected the integral to go to infinity due to low temperature limits.
Yes. The Eq. 8 is any temperature, but to get to low T, we take the integral to infinity of course.