We then *immediately* get the following resultspic.twitter.com/rvFxC5TDev
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Isn't it more straightforward to derive the same through atan's derivative? atan(x) = ∫ˣ₀ 1/(1+t²) dt
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Oh how great our vision is in hindsight! In a way, yes.. If you’re clever enough to come up with trig sub before it was invented.. or you knew the derivative or arctan(x) The beauty of this trick is that it doesn’t require anything except knowledge of derivatives
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Very clever, but not very useful. „Given a complicated integrand, start with a simple one and use identities until you reproduce it“ is not a technique that will guide anyone to a solution.
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probably not, but it shows that sometimes messing around with something simple leads to stumbling upon something nontrivial
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This is basically the way to define \tan as the inversion of an integral. Something first done by Gauss in 1799 when he studied "lemniscate functions". Earlier on the topic:https://twitter.com/ValFadeev/status/1102281091187175424 …
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very cool
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Inserting a 1 is always productive.
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