What are the conditions regarding the behavior of some function as x goes to plus or minus infinity?Well in that case, we are in R so f^2 will be positive so could be too big as x goes to inf. :/
Hi,It's easy to show that (sin(1/x)/x)^2 is not integrable over R because its antiderivative has a simple representation that blows up at a certain point. Hint:
[tex](\sin a)^2 = \frac{1-\cos 2a}2[/tex]
It is showing that sin(1/x)/x is integrable over R that is a bit tougher because this function is not integrable in the elementary functions. It's integral is closely related to a well-known special function. Hint:
[tex]\text{Si}(x) \equiv \int_0^x \frac{\sin t}t\,dt[/tex]
Si(x), the sine integral, is not quite what you want, but it is very close.
what is the range of integration?Give an example of a function f : R -> R, such that f is integrable but f^2 is not
integrable. Prove your result.
Seriously I dont know where to start, please help me guys
I believe the range doesn't matter, so long as f is integrable over the range while f^2 is notwhat is the range of integration?
Well, the particular example does. Some functions are integrable on [itex][0, \infty)[/itex], but are not on [itex](-\infty,\infty)[/itex]. Since you asked for a particular example, I would think it matters. Although, now I see that you have the provision [itex]f: \mathbb{R} \rightarrow \mathbb{R}[/itex], which would mean it is defined on the whole real axis.I believe the range doesn't matter, so long as f is integrable over the range while f^2 is not
[tex]what if we define the range to be [0,1]? still need a clearer hint on a simple f(x).