How to solve integrals like Ramanujan

As an alternative, you could possibly do what Ramanujan himself is reputed to have done, which is to take the book by Carr: "Formulas and theorems in pure mathematics" (it is probably available on Google books), and work your way through the book trying to prove all the results that Carr quotes without proof, and do this in parallel with reading "The Mathematical Legacy of Srinivasa Ramanujan" by Murty and Murty.

Unfortunately, Ramanujan seems to have been rather poor writing out formal proofs of the amazing results he obtained, so I am not sure that my idea would be a very productive way to proceed, but it might possibly provide some insight into his way of making mathematical discoveries.

Look up "Ramanujan's Notebooks" in both Google and Amazon. In Amazon, you will get a chance to spend a lot to learn a lot. In Google, you will get links like this: http://www.plouffe.fr/simon/math/Ramanujan's%20Notebooks%20I.pdf. [2017-12-13: this link is not valid today.]