Bharat is original name of India, as set down in Article 1 of the Constitution, adopted in 1950, which states in English: “India, that is Bharat,…”[20] The term ‘Bharat’ with its origins in Sanskrit, holds a significant historical and cultural significance. Throughout India’s ancient history, the name ‘Bharat’ has been linked to various legends and events, contributing to its enduring narrative.[21]

According to ancient texts and scriptures, one of the earliest references to the name ‘Bharat’ dates back to the Rig Veda’s accounts of the legendary ‘Dasharajna’ or the Battle of Ten Kings. This epic confrontation involved King Sudasa of the Bharata tribe from the Trtsu Dynasty, ultimately leading to the tribe’s recognition and the land being termed ‘Bharat Varsha’ – the land of Bharata.

The Sanskrit word Bhārata is a vrddhi derivation of Bharata, which was originally an epithet of Agni. The term is a verbal noun of the Sanskrit root bhr-, “to bear/to carry”, with a literal meaning of to be maintained (of fire). The root bhr is cognate with the English verb to bear and Latin ferō. This term also means “one who is engaged in search for knowledge”.

This realm of Bharat, which has been referred to as Bhāratavarṣa in puranas - after Bharata, the son of Rishabha.

The country that lies north of the ocean and south of the snowy mountains is called Bhāratam there dwell the descendants of Bharata.

\—Vishnu Purana (2,3,1)

In the Sanskrit epic, the Mahabharat (200 BCE to 300 CE), a larger region of Indosphere is encompassed by the term Bharat.[28] Some other Puranic passages refer to the same Bhārata people, who are described as the descendants of Dushyanta’s son Bharata in the Mahabharata.

https://en.wikipedia.org/wiki/Names_for_India#Bharat

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In case you like 90s Bollywood, here’s a song (music debut by AR Rahman): https://www.youtube.com/watch?v=7q5DUIgLs_4