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I was browsing through Carr's Synopsis, the book made famous by Srinivasa Ramanujan (available in archive.org), in page 319 . I found this: enter image description here

We usually integrate only definite integrals term by term, subject to some conditions. Are there conditions for indefinite integrals also? This is a problem in differential algebras, not calculus, I suppose.

S. Venkataraman
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    You should be inside the radius of convergence of the power series. Of course the indefinite integral formula means exactly that the derivative of the series on the right is the integrand. All modern calculus texts should discuss this. – GEdgar Dec 21 '23 at 03:43
  • Would you please, for some of us, snopsise the snopsis referred to? – A rural reader Dec 21 '23 at 04:35
  • @ A rural Reader Hi! Sorry for the typo. It is actually Synopsis. This refers to the book `Elementary Results in Pure Mathematics containing Propositions, Formulae and Methods of analysis with abridged demonstrations' by G. S. Carr, commonly called Carr's Synopsis. The self taught Indian Mathematician Srinivasa Ramanujan learnt most of Mathematics from this book. – S. Venkataraman Dec 21 '23 at 08:21
  • The power series resulting from the integration will always converge in the same region the original power series did, and to a function whose derivative will be the original power series. However, if the power series involves terms with negative exponent as in the example, that region will not include $0$, but will be symmetric about $0$ (remember that $\int \frac{dx}x = \log|x| + C$, not just $\log x + C$). Further, the "$C$" in the two regions of convergence can have different values. But the FTC only applies over a single connected region. – Paul Sinclair Dec 22 '23 at 19:41

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