This isn't really a homework question, more just something I noticed while evaluating an integral and was curious about: At this stage, I was able to simplify the expression before solving for the integral algebraically (since the second iteration yielded the original integral the right side) and in doing so I came to the correct answer. I noticed however that I could have kept applying integration by parts over and over again. Had I done this, would it be possible to evaluate the integral by applying some property of series to the coefficients to see if they converge at some value? I know that the best way to solve the integral is the method I ended up using, but I was just curious if there were some relationship between integrals, coefficients, and series. I only have a conceptual understanding of series at this point so pardon me if my question isn't a valid one. Thank you.