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I know that for example 1/x^(1/2) would work for (0,1), but are there any examples on [0, ∞)?

Also, any examples of these functions where you can actually calculate the exact value of ∫f(x)dx over that [0, ∞)?

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Yes there are. Think of a function which is zero everywhere, except at intervals around the points $(n,0)$, where its graph consists of the sides of an isosceles triangle centered at the point $(n,0)$ with area $1/n^2$, and height $n$. Here $n$ denotes a natural number.

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$f(x) = x(\sin^2 x)^{x^5}$ has this property. But the building block approach in the answer of @uniquesolution is more intuitive, and there the integral can be computed exactly.

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