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The infinitive form of a verb is created by adding "to" + the verb. For example "to teach" or "to read." What is important about infinitives that they are not the action or linking verb of the sentence, they actual serve as other parts of speech in the sentence -- they can be subjects, direct objects, predicate nouns, adjectives or adverbs!
In the sentence from your post, you could make the sentence more concise by writing:
He hopes to know by tomorrow.
In this sentence "He" is the subject, "hopes" is the main/conjugated verb, "to know" is the direct object. By writing it this way you avoid the unnecessary repetition of he and the dependent clause (that he will know).
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