<s>[INST] Why can't we integrate $\int\frac{1}{1+x^2}$ to $\frac{\ln(1+x^2)}{2x}$? $\int\frac{1}{1+2x}$ is integrated to $\frac{\ln(1+2x)}{2}$. As we are multiplying and dividing by the derivative of the function inside, why is it not possible to do the same with non-linear polynomials. For example, why can't we do the following:
$\int\frac{1}{1+x^2}$ = $\frac{\ln(1+x^2)}{2x}$
 [/INST] Because
$\frac{d}{dx}\frac{\ln(1+x^2)}{2x}=
\frac{1}{x^2+1}-\frac{\ln(x^2+1)}{2 x^2}$
And as you know:
$\frac{1}{x^2+1}-\frac{\ln(x^2+1)}{2 x^2}\neq\frac{1}{x^2+1}$
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