# Example of a Lebesgue integrable function over [0, ∞), that is countiuous and not bounded [closed]

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, ∞)?

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.

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