Digital integration

Calculus Level pending

Define the function $$f(x)$$ on $$(0,1)$$ as the one that takes the standard decimal representation of $$x$$ and swaps all odd and even indices, and then interprets the resulting sequence of digits as the decimal representation of another real number: $f(0.x_1x_2x_3x_4\ldots) = 0.x_2x_1x_4x_3\ldots.$ Now define $$g(x)$$ on $$(0,1)$$ as a function that takes each digit $$x_n$$ in the decimal representation of $$x$$ and replaces it with $$7x_n + 2 \pmod{10},$$ and then similarly interprets the result as a new real number.

Find the integral of $$h(x)=g\big(f(x)\big)$$ over its entire domain.

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