1. He didn't even bother to let me know he was coming.
2. He didn't even bother letting me know he was coming.
(Can we use both sentences? Do we put 'to-infinitive' and 'gerund' after 'bother?)
They both mean the same thing, but the first has the better construction: "bother" + an infinitive.
Yes, both sentences are grammatically correct and convey the same meaning. In these sentences, "bother" is used as a transitive verb, meaning it is followed by an object and can be followed by either a to-infinitive or a gerund.
In the first sentence, the to-infinitive form "to let" is used after "bother," and the pronoun "me" serves as the object. So, the sentence structure is: subject (He) + didn't + even bother + to let + object (me) + know + object complement (he was coming).
In the second sentence, the gerund form "letting" is used after "bother," and again, "me" is the object. The sentence structure is: subject (He) + didn't + even bother + gerund (letting) + object (me) + know + object complement (he was coming).
Both sentences are equally correct, and the choice between the to-infinitive and gerund form depends on personal preference or style.