Having almost finished an honours calculus sequence, I'm now about to enroll in a real analysis course. I've been buying some interesting books lately on top of my required reading list (for example, I supplemented my Calculus book, which was Peter Lax' "Calculus With Applications", with Mercer's "More Calculus of a Single Variable"), and I'm now faced with the opportunity to buy a book on various Calculus techniques and tricks - Klambauer's "Aspects of Calculus", to be precise - but I'm not sure if working through this book will be worth my time, or if I'd be better served spending my spare time working through some other topic.

So, in trying to find an answer to this question, I think it's fair to generalize it and ask instead, how important are "advanced" differentiation and integration techniques and tricks to a mathematics major intending to work towards a PhD in mathematics - possibly aspiring to work in academia? I'd imagine such a question leads to different answers depending on what area of mathematics you decide to work in, so I'll leave the area unspecified.

Also, I'm aware this question is somewhat similar to this one, but I'm specifically asking about the advanced techniques and tricks as discussed in Klambauer's book, not so much about calculus in general. Klambauer's book is quite particular about what it covers (difficult rational integration examples, hard derivatives, complicated series etc etc).

Edit: again, allow me to emphasize that the answers will depend on what area of mathematics one decides to work in. I'm aware of that, and I knowingly and purposefully left the mathematical area unspecified so as to encourage open-ended answers that will end up painting a more complete picture than only focusing on the relevance of advanced differentiation tricks and techniques in just algebraic geometry, for example.

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    $\begingroup$ It entirely depends on which kind of mathematician you would like to become. For a general topologist, to know how to compute a series through creative telescoping and the inverse Laplace transform might be irrelevant, for a number theorist is essential. So, very opinion-based, and the opinion that matters the most is yours. $\endgroup$ – Jack D'Aurizio Dec 15 '16 at 14:40
  • $\begingroup$ I am dubious of the notion that leaving "the mathematical area unspecified" would "end up painting a more complete picture". In many ways this is not the right place to try and paint a complete picture of the value of "differentiation and integration tricks". Such techniques have been built up over hundreds of years at this point. Please review How to Ask. $\endgroup$ – hardmath Dec 15 '16 at 23:05
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    $\begingroup$ @hardmath A strange answer, bordering on rudeness. I left the mathematical area unspecified precisely because having a topologist say one thing and an algebraic geometrist say another gives me a clearer idea of what to expect, and whether to pursue learning about advanced integration and differentiation techniques not commonly taught at my stage of mathematics education. Saying that "this is not the right place" for somebody to discuss this and learn about its relevance to one's mathematics education verges on being presumptuous. $\endgroup$ – Ius Klesar Dec 15 '16 at 23:31
  • $\begingroup$ The comment I left above is not an answer and is intended not as a merely rude response but as constructive criticism. You can ignore it if it is not helpful in understanding my vote to close this Question. $\endgroup$ – hardmath Dec 15 '16 at 23:35
  • $\begingroup$ May I suggest that your "open-ended answers" are better sought in the chatrooms than in the Q&A portion of Math.SE? It was not for nothing that I linked to the "How to Ask" FAQ. If you invited me to a chat, I'd be happy to join you there. $\endgroup$ – hardmath Dec 15 '16 at 23:40

I would argue that if you are an undergraduate who is planning to obtain a PhD in math, then your time is better spent taking a set of math classes as diverse as possible. If you do end up specializing in a field that involves heavy use of advanced integration trickeries, then you would still have 5 years during your PhD to learn that anyway.

Also, maybe take a complex analysis class and learn a bit of contour integration first, before delving into advanced integration. A class in numerical techniques could be very interesting change of perspective as well.

I guess what I am saying is try to learn all the basic methods of doing integrals first, perhaps then maybe decide if you want to specialize in it.

But if you really enjoy doing tricky integrals no matter what, (like me with solving inequalities), then there is no harm in devoting extra time working through those texts.

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