
ergeben
to show / yield
To have as a result or consequence; to lead to something.
Die Untersuchung ergab, dass die Ursache ein technischer Defekt war.
The investigation showed that the cause was a technical defect.
to result / surrender
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to show / yield
To have as a result or consequence; to lead to something.
Die Untersuchung ergab, dass die Ursache ein technischer Defekt war.
The investigation showed that the cause was a technical defect.

to surrender
To give up the fight and submit to the opponent; to capitulate.
Die belagerte Stadt ergab sich nach wochenlangem Widerstand.
The besieged city surrendered after weeks of resistance.

to arise from
To follow logically or become known from a situation or information.
Aus den vorliegenden Daten ergibt sich ein klarer Trend.
A clear trend arises from the available data.

to resign oneself
To submit without resistance to an unavoidable situation or fate.
Nach der Diagnose ergab er sich in sein Schicksal.
After the diagnosis, he resigned himself to his fate.
The prefix 'er-' is inseparable. It stays firmly attached to the verb 'geben' (to give) in all sentence forms and is not placed at the end of the sentence.
Die Analyse ergab ein überraschendes Resultat.
The analysis yielded a surprising result.
This verb can be used reflexively ('sich ergeben') or non-reflexively ('ergeben'). Without 'sich', it means 'to have a result'. With 'sich', it means 'to surrender' or 'to resign oneself'.
Zwei plus zwei ergibt vier.
Two plus two equals four.
In the meaning 'to have as a result', the subject is often a word that describes a process, such as 'die Studie' (the study), 'die Berechnung' (the calculation), 'die Suche' (the search), or 'die Umfrage' (the survey).
Die Berechnung ergab eine Summe von über 1000 Euro.
The calculation yielded a total of over 1,000 euros.
When something follows logically from information, the fixed construction 'sich ergeben aus' + dative is used. The new information is the subject of the sentence.
Aus dem Gespräch ergab sich die Notwendigkeit für ein weiteres Treffen.
The conversation showed the need for another meeting.
When 'sich ergeben' means 'to surrender', the person or group one submits to is often in the dative. Typical examples are 'dem Feind' (to the enemy) or 'der Polizei' (to the police).
Nach einer langen Verfolgung ergab sich der Dieb der Polizei.
After a long chase, the thief surrendered to the police.
In the meaning 'to resign oneself', the fixed expression 'sich in sein Schicksal ergeben' (to resign oneself to one's fate) is often used. Here, 'in' takes the accusative.
Er war sehr traurig, aber schließlich ergab er sich in sein Schicksal.
He was very sad, but eventually he resigned himself to his fate.
to give
Both geben and ergeben can name the result of a calculation. Geben sounds more informal, while ergeben is the more precise standard-language form for a result or consequence. In a factual mathematical statement, ergeben is therefore usually the better choice, although geben is also possible in conversation.

Zwei plus drei ergeben als Ergebnis fünf.
Two plus three equals five.
Zwei plus drei gibt als Ergebnis fünf.
Two plus three equals five.