What does encryption with RSA public key provide? I want to know if encryption with RSA public key provides confidentiality and integrity. I'm quite sure that it provides confidentiality but I have concerns about integrity. 
If an attacker would scramble ciphertext what would be the result I get?
Do I need to put some kind of MAC at the end of messages before encryption?

If I have to be more specific about my problem, I want to use RSA for sending tiny requests to a server. I am thinking of encrypting requests with server's public key. Server will also know my public key and send back responses encrypted with my public key. I'm not sure if I'm using these correctly since I'm a beginner in these topics.

Any help would be appreciated.
Properly done encryption with the RSA public key of the server provides confidentiality of the request, but not integrity or proof of origin (anyone can forge a request with any desired content). 

There must be some secret on the sender's side to prove integrity. That can be an RSA private key used for RSA signature (with sender public keys known to the server either by transmission over a channel with integrity, or by way of a digital certificate). Other signature schemes also work (DSA, ECDSA..).

Symmetric crypto also works for integrity, if it is acceptable that the server holds the client's secrets (or a master key allowing the server to re-compute the  secrets of all clients).

Symmetric crypto can provide both integrity and confidentiality (e.g. Salsa20-Poly1305 per RFC&nbsp;7539), but asymmetric encryption has a superiority: the confidentiality that it offers does not depend on the client's ability to keep a key secret on the long run; only integrity does.