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What are the restrictions on using substitution in integration?

For example, integration in this case may be done by the substitution $u=\tan x$, as seen in the following $\textit{Mathematica}$ notebook

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Integration done by substitution $u=\tan {x\over 2}$.

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The source function is a continuous positive function over $\mathbb {R}$ which implied by necessity that its antiderivative is continuous and increasing over $\mathbb {R}$ (which contradicts the graph of its analytical antiderivative that has jumb discontinuities).

When I sent this question to Dr.Math, he replied - thanks to him - that the antiderivative of the function $ f(x)={1\over {1+{\sin x}^2}} $ is:

$$ {\tan^{-1} (\sqrt 2 \tan x) \over \sqrt 2} + \lfloor \frac x \pi-\frac 12 \rfloor * \frac{\pi}{\sqrt 2} + C $$

Which matches the "True" antiderivative of the function after redefining the integral at $( {\pi \over 2} + n.\pi) $. The second term has a derivative of $0$ over $ \Bbb R $.

This kind of "weird" functions makes me wonder about the "Restrictions" on using substitution when evaluating integrals, how to know that the function I work on needs refinements, and how to construct this "True" antiderivative ?

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  • $\begingroup$ This site is not designed for "open discussion"---perhaps you could replace that statement with a concrete question? $\endgroup$ Commented Jul 10, 2015 at 15:29
  • $\begingroup$ statement removed $\endgroup$ Commented Jul 10, 2015 at 15:33
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    $\begingroup$ David J. Jeffreys has written several articles about this. Try googling "The importance of being continuous", for example. $\endgroup$ Commented Jul 10, 2015 at 15:53
  • $\begingroup$ @HansLundmark Thank you very much, I googled and found this paper and it's exactly what I was looking for. Thanks a lot. :) $\endgroup$ Commented Jul 10, 2015 at 16:11
  • $\begingroup$ You're welcome! $\endgroup$ Commented Jul 11, 2015 at 5:59

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