What is the difference between analysieren and untersuchen?
- Ich untersuche den Fall Nicole Smith.
- Ich analysiere den Fall Nicole Smith.
I think there is only a subtle difference in that "analysieren" tends to be usable only when mainly mental work is done.
It would sound funny if you say:
Ich analysiere das defekte Getriebe.
Der Doktor untersuchte mich. Er nahm mir Blut ab, das im Labor analysiert wurde.
But in your case, you can use both forms.
If you are unsure, prefer "untersuchen", it is always valid, while "analysieren" is not.
Depending on context both "analysieren", and "untersuchen" may be used as synonyms but this is by far not always the case.
To get a grasp of the subtle differences in meaning and usage it helps to look at the etymology of "Analyse":
"resolution of anything complex into simple elements" (opposite of synthesis), from Medieval Latin analysis (15c.), from Greek ἀνάλυσις "a breaking up, a loosening, releasing," Online Etymology Dictionary
This implies that whenever we look at dissected details of anything on examination ("Untersuchung") we may use the verb "analysieren".
This usage is the same in German than in English and can e.g. be the case of a laboratory analysis, a forensic analysis, or other any other analysis of single aspects. Then we may use both, "untersuchen", or "analysieren", but we do not use "analysieren" in the case where we look at something as a whole.
They're both the same, semantically. At least, I'd be hard pressed to find examples where you couldn't technically use one for the other.
The difference is that »untersuchen« is easily understood by anybody whereas »analysieren« is Greek and so sounds more "scientific" and "important" and may not be understood by all children.
I think the best translation for »untersuchen« is »investigate«, and literally »underseek« (never seen that in English, but there's »onderzoeken« in Dutch). So it's seek and search, if boiled down to simple verbs.
»Analyze«, on the other hand, is »dissolve« in Latin (and English), and the proper German translation is »auflösen«, like »Zucker in Wasser auflösen« oder »ein Geheimnis/Rätsel/Mysterium auflösen« (oder auch »lösen«).
So, »untersuchen« is more down-to-earch, and »analysieren« is more like in a chemical lab, if you look at what kind of attitude the word conveys.
Regarding your example: »einen Fall untersuchen« is better, no question about it. Einen Fall untersuchen. Eine Probe untersuchen (Bodenprobe, Gewebeprobe, Blutprobe).
There is no general answer to this question. It always depends on the context.
Both words can describe an action very specifically:
Wir untersuchen die Funktion f(x)=sin(x).
Wir analysieren das Gedicht.
Both words can describe an action in a broad sense:
Wir untersuchen die Gründe für den Wahlsieg der CDU.
Wir analysieren die Gründe für den Wahlsieg der CDU.
Sometimes, both words are (a little bit) different: first pair of examples. Sometimes, both words are synonymous with different connotation: second pair of examples.
Sometimes, both words are exactly synonymous:
Wir untersuchen das Spurenmaterial des Täters.
Wir analysieren das Spurenmaterial des Täters.
Sometimes, only one of the two words can be used.
Der Doktor untersucht den Patienten.