Unable to create / Move data to New Partitions from the existing partition SQL Server 2014
If your intention is to partition this data by year/month, I think your current partitioning column (which is an integer and not a 'true' date data type and not even in yyyymmdd format even as an integer) is all wrong for the job.  I'm also assuming you don't have another column in the table that IS the MonthlyDate in a true Date data type.

In my opinion (and assuming you have extra disk space), you should:


Create a new partition function that uses a DATE data type.  It is recommended that this should be RANGE RIGHT (see Dan Guzman's link at the bottom of this answer for explanations).  Make sure you create it with ranges well below your minimum date as well as ranges well above your maximum date.  Also, as time goes by, make sure you are splitting empty partitions well ahead of any data already being populated in them.
Create a new partition scheme that references the new partition function.
Create a new 'replacement' table (that references the newly created partition scheme) by scripting a create of the current table, but add a computed/persisted DATE column (ex:PartitionedColumn DATE) that is derived/converted from your integer MonthlyDate column.
Set up an insert statement, inserting records from the existing table into the new table, say a few thousand or a hundred thousand in a batch, within a transaction. When the transaction completes perform a log backup (if you're not in SIMPLE recovery).  That'll keep you transaction log from running out of space. Then repeat the batch... keep doing that until all records are written to the new table with the persisted computed column.
Then change the old table name, change the new table name to that of the old table


Keep in mind that all unique indexes that participate in a partitioning scenario, MUST have the partitioning column defined as part of the base index definition and not simply an included column.

Also, to enable partition switching (often used in a "sliding window"), all indexes on the table must be aligned.  BOL has good information about this.  Look for Special Guidelines for Partitioned Indexes.

Lastly, check out Dan Guzman's excellent post on Sql Server partitioning.