# What are the restrictions on using substitution in integration?

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

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