# Riemann Lebesgue integrals

• Jul 1st 2012, 11:59 AM
kezman
Riemann Lebesgue integrals
Im having some trouble with these ones

Decide if the following integrals exist as Lebesgue integrals and as Riemann integrals

$\int_{1}^{+ \infty} \frac{dx}{x^2}$

$\int_{0}^{+ \infty} \frac{sen^2(x)dx}{x^2}$

Thanks!!!
• Jul 1st 2012, 12:53 PM
Plato
Re: Riemann Lebesgue integrals
Quote:

Originally Posted by kezman
Decide if the following integrals exist as Lebesgue integrals and as Riemann integrals

$\int_{1}^{+ \infty} \frac{dx}{x^2}$

$\int_{0}^{+ \infty} \frac{sen^2(x)dx}{x^2}$

Are they Riemann integrable? If not why not?
• Jul 2nd 2012, 07:59 AM
kezman
Re: Riemann Lebesgue integrals
As for riemann integral the first one exist and is = -1
The second one doesnt exist.

The problem is to prove if exists the lebesgue integral, I have no clue.:(
• Jul 2nd 2012, 08:25 AM
Plato
Re: Riemann Lebesgue integrals
Quote:

Originally Posted by kezman
As for riemann integral the first one exist and is = -1
The second one doesnt exist.
The problem is to prove if exists the lebesgue integral, I have no clue.:(

The value of the first is 1, not -1.
For the second did you notice that $\frac{\sin^2(x)}{x^2}\le\frac{1}{x^2}$.
• Jul 2nd 2012, 12:05 PM
kezman
Re: Riemann Lebesgue integrals
ok the second integral if exists must be between 0 and 1
• Jul 2nd 2012, 01:01 PM
Plato
Re: Riemann Lebesgue integrals
Quote:

Originally Posted by kezman
ok the second integral if exists must be between 0 and 1

Here is a reference for you.

And another.
• Jul 2nd 2012, 02:00 PM
kezman
Re: Riemann Lebesgue integrals
Excelent, so Ill have to fight against lebesgue integral.