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Take a look at the integral - $\int \frac{1}{1+\sin^2x}$. The integrand is defined for every $x\in \mathbb{R}$

$\\$ most solutions use the substitution $t = \tan(x)$. $\\$

my question is what about $x=\frac{\pi}{2}+\pi \cdot k$ ? $\\$

the substituion is not defined, and the solution is not defined in these points.

mcr0yal
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  • See comments and linked questions here: https://math.stackexchange.com/questions/1356523/what-are-the-restrictions-on-using-substitution-in-integration – Hans Lundmark Sep 06 '21 at 14:31
  • the substitution can be applied just in the regions where the integrand is well-defined. However, in the critical points, we can understand the integral as improper – Masacroso Sep 06 '21 at 14:35

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