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Tomorrow, I will have a test about Calculus 1 and 2 and up to now, I was solving exercises over and over again but I have the impression that I don't learn a lot from doing this. I believe that when I've already solved a similar integral before, I know how I have to substitute - whereas whenever I'm confronted with an integral I've never seen before, I have problems finding the correct substitution to solve it. The same applies to infinite series.

Thus, I wanted to ask whether there are some standard substitutions one uses over and over again or some standard techniques to find the value of an infinite series (or to show that it diverges). I already know some tricks but I rarely realize when I have to apply them (mostly for integral substitutions). E.g. sometimes my assistant professor gave us the hint to substitute $t = \tan(\frac{x}{2})$. I did that and it worked out perfectly well, however I now have no idea when I have to use this substitution again.

So to make it short: Can anyone give/link a list of common integral substitutions and techniques to evaluate infinite series?

Thanks for any help in advance.

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We should probably make this a community wiki, since you are looking for many suggestions. My advice: always know the general methods for certain types of standard integrals. For example, learn how to integrate rational functions in a completely straight forward way. Then learn that $t=\tan(x/2)$ converts integrals of a rational function of $sin x, cos x$ into a rational function of $t$. It's much better to know the general methods than some lucky tricks, unless those tricks happen to come up enough in what you do. – Ragib Zaman Aug 25 '11 at 8:51
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Exactly those general methods are what I am asking for. I already realized that some substitutions can be used more often than other ones (at least for me) but I think I do not know a lot of the general methods already and I also don't always know when to apply them. – Huy Aug 25 '11 at 9:00
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For the general methods, consult any book on Integral Calculus (I suggest Apostol, Vol 1). – Ragib Zaman Aug 25 '11 at 9:25
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I agree with Ragib. What you're looking for ought to be in the textbook for your calculus course. If they're not, go to the library and find a better book. ;-) – Hans Lundmark Aug 25 '11 at 10:09
This solves my problem regarding integrals (assuming I really find such a textbook - mine does not contain a lot of typical substitutions); but I'm still looking for tricks to evaluate infinite series. I know some criteria to check for convergence but sometimes, the solution is much easier, e.g. they use the Taylor expansion. – Huy Aug 25 '11 at 11:37
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