What term is used to describe the illicit funds that enter the banking system undetected and appear legitimate?

The term used to describe illicit funds that enter the banking system undetected and appear legitimate is "integrated." In the context of money laundering, integration refers to the final stage in the money laundering process, where the illicit funds are reintroduced into the economy in such a way that they appear to be legitimate income. At this stage, the money has gone through various transactions designed to obscure its illegal origins, making it difficult to trace back to criminal activities.

In contrast, the other terms relate to different stages or methods of money laundering. Structured typically refers to the initial stage where large amounts of cash are broken down into smaller amounts to avoid detection. Layered denotes the process of concealing the source of the funds through complex layers of transactions, often involving multiple accounts or banks. The phrase "properly washed," while it might suggest a cleaning process, is not an industry standard term used in money laundering vocabulary. Thus, the term integrated accurately reflects the culmination of successfully laundering illicit funds, aligning perfectly with the context of the question.