Infinitival clauses, also known as infinitive phrases, are constructions that involve the preposition “zu” plus an infinitive form of a verb, and they typically occur at the ends of sentences.

Ich habe keine Zeit zu schlafen. 
I have no time to sleep.

In English, whenever there are modifiers or objects associated with the phrases, they usually are found after the infinitive phrase: I have no time to sleep today before work. This same sentence in German would look like this: Ich habe heute vor der Arbeit keine Zeit zu schlafen. As can be seen, the German infinitive phrase occurs after modifiers or objects.

When a separable prefix verb is used with an infinitive phrase, the “zu” is placed between the prefix and the rest of the verb:

Ich habe heute keine Zeit auszuschlafen.
I have no time to sleep in today.

When a modal verb is used in a sentence with an infinitive phrase, the infinitive is placed directly before the “zu,” followed by the modal verb immediately after the “zu.” This is found more often in combination with conjunctions (um, ohne, anstatt, statt).

Ohne die Notwendigkeit schlafen zu müssen, könnte man viel mehr machen.
Without the necessity to have to sleep, one could do much more.

Some infinitive phrases use prepositions to inflect the meanings of sentences. When “um” is added, a purpose or intention is expressed (i.e. “um…zu” = “in order to x”):

Um ein Haus zu kaufen, muss man viel Geld sparen.
/ In order to buy a house, one has to save a lot of money.

The preposition “ohne” is used to connote doing something without something else:

Ohne zu denken, springt der Mann ins Wasser, um das Baby zu retten.
Without thinking, the man jumps into the water in order to rescue the baby.

The prepositions “statt” and “anstatt,” like “ohne,” are translated using the gerund form in English:

Er spielt Videospiele anstatt zur Arbeit zu gehen.
He plays video games instead of going to work.

Wir gehen heute Abend ins Theater statt in die Oper zu gehen.
We are going this evening to the theater instead of going to the opera.

Notice that in each of the above examples, the verb “to go” is written as the gerund “going” after the prepositions “ohne,” “anstatt,” and “statt.”