
sich bemühen
to strive
To make an effort or try hard in order to achieve a specific goal.
Sie bemüht sich sehr, ihre Deutschkenntnisse zu verbessern.
She is trying very hard to improve her German skills.
to strive / to trouble
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to strive
To make an effort or try hard in order to achieve a specific goal.
Sie bemüht sich sehr, ihre Deutschkenntnisse zu verbessern.
She is trying very hard to improve her German skills.

to trouble someone
To ask someone for help or a service and thereby inconvenience them. Usually used in formal or polite contexts.
Ich möchte nicht extra den Hausmeister bemühen, die Glühbirne zu wechseln.
I don't want to trouble the caretaker just to change the light bulb.

to invoke something
To draw on or use something, for example an example, a comparison, or a quote, in order to explain or support something.
Um die komplexe Situation zu erklären, musste der Redner einen historischen Vergleich bemühen.
To explain the complex situation, the speaker had to invoke a historical comparison.
'Bemühen' (to strive / to trouble) is related to the noun 'die Mühe' (effort). The inseparable prefix 'be-' turns it into a verb that describes an action.
Er bemüht sich, pünktlich zu sein.
He is trying to be on time.
The meaning depends on the object: Is it used with a reflexive pronoun ('sich bemühen' — to make an effort), a person ('jemanden bemühen' — to trouble someone), or a thing ('einen Vergleich bemühen' — to invoke a comparison)?
Ich bemühe mich um eine schnelle Antwort.
I am making an effort to provide a quick answer.
In this reflexive meaning, there are two typical constructions: 'sich bemühen um + Akkusativ' (to make an effort to get/obtain something) or 'sich bemühen, etwas zu tun' (to try to do something; infinitive with zu).
Der Student bemüht sich um ein Stipendium.
The student is trying to get a scholarship.
The expression 'jemanden bemühen' (to trouble someone) is a very polite and formal way to ask someone for help. It expresses that you are aware of the inconvenience caused.
Dafür müssen Sie nicht extra den Chef bemühen, das kann ich für Sie erledigen.
You don't need to trouble the boss especially for that; I can take care of it for you.
This meaning is often used in argumentative or intellectual contexts. Typical objects are: einen Vergleich bemühen (to invoke a comparison), ein Beispiel bemühen (to invoke/use an example), eine Statistik bemühen (to cite/use a statistic).
Um seinen Standpunkt zu stärken, bemühte er eine aktuelle Studie.
To strengthen his point of view, he cited a recent study.
to try / taste
Versuchen and sich bemühen can both express that a goal has not yet been reached for certain. Versuchen emphasizes the individual or general attempt, while sich bemühen highlights the willingness and effort involved. Choose sich bemühen when the person’s effort is particularly important.

Ich bemühe mich, die schwere Tür mit viel Kraft zu öffnen.
I’m making a great effort to open the heavy door.
Ich versuche, die schwere Tür zu öffnen.
I’m trying to open the heavy door.