Currently bittrex can find out if you reused a monero address if you used the same one. Can one use an integrated for bittrex withdraws? integrated addresses is the public key and the payment id combined. But is it still a "real" address where you can use it in replacement of giving out your main monero address?
An integrated address is a valid address, and if the exchange has implemented the support for integrated addresses, you can use them as destination when withdrawing from the exchange.
However, the basic monero address and the payment ID can be recovered from an integrated address (integrated addresses are just a way to pack an address and a payment ID together, nothing is encrypted). Therefore whether you use a basic address or an integrated address (for that basic address) to withdraw, the exchange will know that these withdrawals have the same wallet as destination.
Using different sub-addresses for each withdrawal can hide the fact that they have the same wallet as destination.