Page 288 - Practical English Usage 3ed - Michael Swan, Oxford

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The marker to is normally used before infinitives (e.g. He wanted to go). Note that this to is not a preposition; after the preposition to we use -ing forms (see 298.2). For infinitives without to (e.g. She let him go), see 281.
split infinitive
A 'split infinitive' is a structure in which to is separated from the rest of the infinitive by an adverb.
I'd like to really understand philosophy.
He began to slowly get up offthe floor.
Split infinitive structures are quite common in English, especially in an informal style. Some people consider them incorrect or careless, and avoid them if possible by putting the adverb in another position.
He began slowly to get up off the floor.
For details of the use of infinitives, see the foUowing sections.
For the use of to instead of a whole infinitive (e.g. I'd like to), see 182.
infinitives (3): without to
We usually put to before the infinitive (e.g. I want to know, It's nice to see you).
But we use the infinitive without to in some cases. after modal auxiliary verbs
After the modal auxiliary verbs will, shall, would, should, can, could, may, might and must, we use the infinitive without to.
I must go now. (NOT J mftSt ttJ go ntlw.)
Can you help me? Do you think she might be joking? I would rather go alone. She will probably be elected.
We also use the infinitive without to after had better (see 230), and sometimes after need and dare (see 366, 151).
You'd better see what she wants.
She needn't do the washing up. I daren't go out at night. . The to-infinitive is used after ought (see 403).
after let, make, hear etc
Certain verbs are followed by object + infinitive without to. They include let, make, see, hear, feel, watch and notice.
She lets her children stay up very late. (NOT She lets her ehiltlFen ttl attiy. .. OR She lets her ehiltEre", attiying ...)
I made them give me the money back.
I didn't see you come in.
We both heard him say that I was leaving. Did you feel the earth move?
Help can also be used in this way (see 244). Could you help me (to) unload the car?
This structure is also possible with have (see 238) and know (see 313). Have Mrs Hansen come in, please. (especially ArnE)
I've never known him (to) pay for a drink.
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infinitives (3): without to 281
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