Hi Ninu,
Can you advise what you mean when saying that you deployed it forcefully? If not using SUM or JSPM (by the way, JSPM has been deprecated in newer releases anyway), did you deploy it using NWDS or AS Java Telnet deploy command? Besides options mentioned above and ANT scripts (all listed in SAP Note 1715441), I don't think there is another alternative to correctly deploy these libraries in a SAP supported way.
Normally, when com.sap.aii.adapter.lib.sda is deployed correctly, provider.xml shall not appear in /lib directory on the application server - /lib directory shall only contain referenced JAR files of JDBC/JMS drivers. Having this in mind and looking into screenshot of tx. AL11 that you provided, I would assume MySQL JDBC driver library and updated provider.xml where copied to corresponding directory on the application server, but not actually consistently deployed as a part of newly assembled com.sap.aii.adapter.lib.sda. Assuming this, can you deploy com.sap.aii.adapter.lib.sda by one of supported means mentioned above and check again if MySQL JDBC library gets listed in Java class loader view?
Can you also post content of file provider.xml contained in deployed com.sap.aii.adapter.lib.sda? Does it have correct references to MySQL JDBC Connector library JAR file in "jars" section?
As a side note, instead of manually assembling com.sap.aii.adapter.lib.sda and changing provider.xml in there, you may give a try to SDA Maker Tool, it helps a lot in automation and simplification of JDBC/JMS and Axis provider libraries assembly (see SAP Note 1987079 for more details on SDA Maker Tool).
Regards,
Vadim