いるのか [Grammar]

[turvy]

In this sentence, what boggles my mind is why you need いるのか if you already have どいう. But I'm no even sure I understand what いるのか means.
Dufourの研究は、虫がどういう生活をしているのかを観察するという方法でした。
Dufour's research talked about his methods to observe how the bugs lived their lives.

 

[Toritoribe]

どういう生活をしているのか is an indirect quotation of a question, as you translated correctly.