Example of a Riemann integrable function, which is not bounded?

Question

I have to give an example of a function which is Riemann integrable on [0,+∞) but its not bounded. I know about the connection of Riemann Integrability and bounded functions, but this is my assignment and I think there is an intention behind it.

I thought about 1/x. We have done the integral of 1/x and the result is +∞. Would that work?

Edit: I understood that the function 1/x would not work. Would maybe the function f(x)=x work?

Answer 1

$\frac 1 {\sqrt x}$ has finite improper Riemann integral on $[0,1]$.

Answer 2

Riemann integral is defined for bounded functions on a finite interval, so neither unbounded functions nor functions on an infinite interval can be Riemann integrated.

On the other hand, you can have an unbounded, improper Riemann integrable function on $[0, +\infty)$, simply by extending Kavi's example from $[0,1]$ to $[0, +\infty)$. For example $$f(x)= \begin{cases} 0 & x= 0 \\ \frac{1}{\sqrt{x}} & 0< x\leqslant1 \\ \frac{1}{x^2} & x > 1 \end{cases} $$ will do the job.

Answer 3

$x \mapsto 1/x$ does not work as the integral

$$\int_0^\infty 1/x dx = +\infty$$

is not finite.