Iíll start with the rare latter two. Anyone who has read the Aeneid
will remember that the imperfect indicative is often written as an infinitive;
exclamabant becomes exclamare. This is the historical infinitive.
An indirect statement, with an infinitive and subject accusative, can stand
alone if the tone be one of exclamation; in English, we would add
"to think thatÖ!", as in: "to think that I should write a paper on Yom
Kippur!"
As a noun the infinitive is incredibly straight forward (e.g., donare bonum est, "it is good to give"), except for two usages. The infinitive can really only be used as an accusative when in apposition to a regular direct object. A&G gives a good example: quam multaÖfacimus cause amicorum, precari ab dignoÖ "How many things we do for our friends, ask favors from an unworthy personÖ" Also, the infinitive almost always accompanies an impersonal verb (the licet, oportet, necesse est crowd) as the predicate nominative of the subject, "it." "It is necessary to eat."
A complentary infinitive aids the main verb in transmitting its meaning. For example, it would not suffice to say "they dare" alone, one must say "they dare to be stupid," etc. Although in Latin the infinitive cannot convey purpose as it can in English, a complementary infinitive can nevertheless be switched with a subjunctive ut clause. This interchangeablility occurs in verbs of willingness, necessity, command, admonition, and others in this vein. This is extended in poetry for artistic effect, like with Horaceís line furit te reperire "he rages to find you."
Infinitive clauses are a type of substantive clause. There are two types of infinitive clauses: those in which the infinitive is interchangeable with the subjunctive, and indirect statements.
Sources:
Jenney, Charles Jr., Rogers V. Scudder, and David D. Coffin.
Third Year Latin (Newton, MA. Allyn and Bacon Inc. 1963)