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If I am not wrong the issue is why displacement work is an inexact differential whereas mechanical work is not always denoted as an exact differential?

Short answer: W=∫Fdx is the generalized equation for mechanical work ,where,

It's most appropriate to write W=∫δw and NOT ∫dw as work is a path dependent process.

So, writing W=∫dw or dw=Fdx is a serious malpractice.

Explanation: You need to specify the properties pressure and volume for a series of quasi static or equilibrium states to specify the path upon which work can be calculated. As work is defined as the area under the curve between the properties P and V in a displacement process one needs to add up all the elemental work for each equilibrium states to get total work done.As there can be more than one quasi static paths possible between two end states of a thermodynamic process different work transfers are possible.Thus work is essentially a path dependent process.

Even when considering mechanical extension or compression of a spring you need intermediate equilibrium positions to be achieved after each oscillation restores to a new equilibrium position.Thus to obtain work using integration we need a series of equilibrium states to specify the path upon which work is to be calculated hence work is practically always an inexact differential.

So it's always appropriate to write δw=Fdx or δw=PdV as it demands the application of mathematical integration to obtain total work.

Hope this helps

Best regards