I know how to check for payments using payment IDs as it is documented on the monero website. But it does not say how to check payments using an integrated address. Can someone tell me how to do this?
An integrated address is just your standard address and the payment id you chose, bound together into one string. When you generated this standard address, you'd have written down that payment id. If not, there is a
split_integrated_address wallet RPC that will do this for you.
That payment id may be used as input for your original RPC call. If it doesn't work, then it's because there's a bug in the RPC that needs fixing, and the best next step is to report it along with the actual commands you used to https://github.com/monero-project/monero/issues
Note that you get a short (64 bit) payment ids when working with integrated addresses, whereas legacy payment ids are 256 bits. This is expected, and you're not missing anything.