`The Story of the Integration of the Indian States` is a book that chronicles the systematic process of integration of princely states into the Indian Union. This book was authored by V.P. Menon (option 4), who served as the Secretary of the States Department and played a crucial role in the political integration of India.

Option 1, B.N. Rau, was a noted Indian civil servant and diplomat, and the Constitutional Adviser to the Constituent Assembly of India, he did not write the mentioned book.

Option 2, C. Rajagopalachari, was an influential leader and thinker in the Indian National Congress. He was the last Governor-General of India. Although a prolific writer, `The Story of the Integration of the Indian States` was not his work.

Option 3, Krishna Menon, was a diplomat and nationalist identified with India`s independence movement. He has written on various subjects but not the mentioned one.

So, the correct answer is V.P. Menon (Option 4). The other options reflect significant figures in the Indian independence era, but none of them wrote `The Story of the Integration of the Indian States`.