# Thread: Riemann Lebesgue integrals

1. ## 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!!!

2. ## Re: Riemann Lebesgue integrals

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?

3. ## 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.

4. ## Re: Riemann Lebesgue integrals

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 $\frac{\sin^2(x)}{x^2}\le\frac{1}{x^2}$.

5. ## Re: Riemann Lebesgue integrals

ok the second integral if exists must be between 0 and 1

6. ## Re: Riemann Lebesgue integrals

Originally Posted by kezman
ok the second integral if exists must be between 0 and 1
Here is a reference for you.

And another.

7. ## Re: Riemann Lebesgue integrals

Excelent, so Ill have to fight against lebesgue integral.