Identify this smooth-cocoon worm from Southern India
This is definitely the larvae of one of a many number of case-bearing moth species. As noted in comments and elsewhere, these case-bearing moths show up fairly frequently on Bio.SE (e.g., see here, here, here, here and here for some examples).  

Most of these moths are members of one of two families: Coleophoridae (the case bearing moths) or Tineidae (the fungus moths).

Some of these species even build cases that they carry around with them. 

Your specific example is possibly Tinea pellionella (case-bearing clothes moth). 

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T. pellionella has a cosmopolitan distribution and occurs nearly worldwide (Cheema 1956). As the Wikipedia article summarizes, the larva feeds mainly on fibrous keratin (e.g., hair and feathers) as well as carpets, furs, upholstery, and woolen fabrics. The larvae remain inside a case it constructs from debris such as hairs and fibers (e.g., from clothing, upholstery, carpets, etc.).