= Solution
Let $\phi:I\to J$ be a $C^1$ <diffeomorphism> between <intervals>, which may be unbounded. For a nonnegative <Lebesgue measurable function> $f$ on $J$, <Lebesgue substitution on an interval> gives
$$
\boxed{\int_J f(y)\,dy=\int_I f(\phi(x))|\phi'(x)|\,dx,}
$$
allowing the value infinity. The same identity applied to $|f|$ shows that $f$ is <Lebesgue integrable> exactly when the transformed integrand is integrable; in that case the displayed formula holds for signed or complex $f$ as well. The absolute value on the <derivative> makes the formula valid for both increasing and decreasing substitutions. Finite endpoints can be omitted without changing a <Lebesgue integral>, because individual points are <null sets>.
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