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There is a strong connection between the result for 1-1 transformation and change of variables in integration?

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When we perform a change of variables in integration, the pdf of the corresponding transformation appears as the integrand.

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$\\int_a^b f(x) dx = \\int_{\\alpha(a)}^{\\alpha(b)}f(\\alpha^{-1}(y))\\Big|\\frac{dx}{dy}\\Big| dy$

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