What are the different types of infinitives? Explain each type with examples.

Infinitives are the base form of a verb, typically preceded by the word "to." They are used to express various functions and can be categorized into different types based on their usage and structure. The different types of infinitives are as follows:

1. Bare Infinitive: This type of infinitive is used without the particle "to" and is commonly used after modal verbs (can, could, may, might, must, shall, should, will, would) and certain other verbs like let, make, help, and have. Examples include:
- She can swim.
- He must leave now.
- Let me go.
- They made him apologize.

2. To-infinitive: This is the most common type of infinitive, where the word "to" is placed before the base form of the verb. It is used in various contexts, such as after adjectives, nouns, and certain verbs. Examples include:
- She is eager to learn.
- I have a desire to travel.
- He wants to become a doctor.
- They decided to go on a vacation.

3. Perfect Infinitive: This type of infinitive is formed by using "to have" followed by the past participle of the verb. It is used to express an action that happened before another action or event. Examples include:
- She seems to have forgotten her keys.
- He appears to have finished his work.
- They claim to have seen a ghost.

4. Continuous Infinitive: This type of infinitive is formed by using "to be" followed by the present participle of the verb. It is used to express an ongoing or continuous action. Examples include:
- She wants to be studying abroad.
- He hopes to be working in a multinational company.
- They plan to be traveling around the world.

5. Passive Infinitive: This type of infinitive is formed by using "to be" followed by the past participle of the verb. It is used to express a passive action or when the subject receives the action. Examples include:
- The car needs to be repaired.
- The report has to be submitted by tomorrow.
- They want the problem to be solved.

6. Split Infinitive: This type of infinitive occurs when an adverb or adverbial phrase is placed between "to" and the base form of the verb. Although it is considered grammatically controversial, it is commonly used in informal English. Examples include:
- She decided to quickly finish her homework.
- He tried to carefully open the box.
- They wanted to completely understand the concept.

In conclusion, the different types of infinitives are bare infinitive, to-infinitive, perfect infinitive, continuous infinitive, passive infinitive, and split infinitive. Each type serves a specific purpose in expressing different aspects of actions or events.