Riemann Lebesgue integrals

**kezman** · July 1st 2012, 10:59 AM

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!!!

**Plato** · July 1st 2012, 11:53 AM

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?

**kezman** · July 2nd 2012, 06:59 AM

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.

**Plato** · July 2nd 2012, 07:25 AM

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

**kezman** · July 2nd 2012, 11:05 AM

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

**Plato** · July 2nd 2012, 12:01 PM

Originally Posted by kezman

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

Here is a reference for you.

And another.

**kezman** · July 2nd 2012, 01:00 PM

Excelent, so Ill have to fight against lebesgue integral.

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