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Fifth Generation (5G) networks are envisioned to be fully autonomous in accordance to the ETSI-defined Zero touch network and Service Management (ZSM) concept. To this end, purpose-specific Machine Learning (ML) models can be used to manage and control physical as well as virtual network resources in a way that is fully compliant to slice Service Level Agreements (SLAs), while also boosting the revenue of the underlying physical network operator(s). This is because specially designed and trained ML models can be both proactive and very effective against slice management issues that can induce significant SLA penalties or runtime costs. However, reaching that point is very challenging. 5G networks will be highly dynamic and complex, offering a large scale of heterogeneous, sophisticated and resource-demanding 5G services as network slices. This raises a need for a well-defined, generic and step-wise roadmap to designing, building and deploying efficient ML models as collaborative components of what can be defined as Cognitive Network and Slice Management (CNSM) 5G systems. To address this need, we take a use case-driven approach to design and present a novel Integrated Methodology for CNSM in virtualized 5G networks based on a concrete eHealth use case, and elaborate on it to derive a generic approach for 5G slice management use cases. The three fundamental components that comprise our proposed methodology include (i) a 5G Cognitive Workflow model that conditions everything from the design up to the final deployment of ML models; (ii) a Four-stage approach to Cognitive Slice Management with an emphasis on anomaly detection; and (iii) a Proactive Control Scheme for the collaboration of different ML models targeting different slice life-cycle management problems.