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Typical courses on real integration spend a lot of time defining the Lebesgue measure and then spend another lot of time defining the integral with respect to a measure. This is sometimes criticized ...

Let $\theta = \tan^{-1}(t)$. Nowadays it is taught: 1º that $$ \frac{d\theta}{dt} = \frac 1{dt\,/\,d\theta} = \frac 1{1+t^2}, \tag1 $$ 2º that, via the fundamental theorem of calculus, this is ...