How to put exercise inside tcolorbox under ExerciseList?

I'm not sure to have understood what you exactly like, but this could be a starting point:

documentclass[a4paper, 11pt]{book}
usepackage{tcolorbox}
usepackage[lastexercise]{exercise}
renewcommand{ExerciseListHeader}{begin{tcolorbox}ExerciseHeaderDifficulty%
textbf{ExerciseListName ExerciseHeaderNB ExerciseHeaderTitle}%
ExerciseHeaderOriginend{tcolorbox}}
BeforeBeginEnvironment{ExerciseList}{vspace{10pt}par}
AfterEndEnvironment{ExerciseList}{vspace{10pt}par}
begin{document}
Something before the list. Something before the list. Something before the list. Something before the list. Something before the list.
begin{ExerciseList}
Exercise Investigate convergence of the following integrals:
Answer An answer.
Exercise Another question:
Answer Another answer.
end{ExerciseList}
Something after the list.
Something after the list.
Something after the list.
Something after the list.
Something after the list.
end{document}