\int_0^{\infty}\frac{\sin x}{x}\ dx

Okay, I'm trying to compute the following integral:

But first, I will try to prove that converge.

.

The integral (Dirichlet's test)

But what about ?

I know that is our *problematic* point and also I know that .

Can I say that because that limit exists the function is integrable?

Now to the the second part-computation.

It's from a book by **G. M. Fikhtengoltz**, *The Fundamentals of Mathematical Analysis*.

(I'm translating this from Russian into Hebrew and then into English.)

We will represent I in form of sum:

.

Putting or and the he says the following:

we substituting or , and we get:

and

Question:

Why we need the both substitutions, __or__ ?

Thank you.

To be continued...

Re: \int_0^{\infty}\frac{\sin x}{x}\ dx

Your integral is just the laplace transformation of evaluated at s=0.

Re: \int_0^{\infty}\frac{\sin x}{x}\ dx

Quote:

Originally Posted by

**General** Your integral is just the laplace transformation of

evaluated at s=0.

Without Laplace transformation please. :)

Re: \int_0^{\infty}\frac{\sin x}{x}\ dx

A more general form of the indefinite integral is...

(1)

This integral has great importance in several applcations, so that almost nobody remembers the 'theoretical bag' hidden in it: the definite integral (1) converges if it is processed as Riemann integral and doesn't converge if it is processed as Lebesgue integral, as for example in...

Lebesgue integration - Wikipedia, the free encyclopedia

All that is well known of course but, proper considering this example, I wonder how to justify motivations regarding the 'superiority' of the Lebesgue integral like that (Thinking)...

*... the Riemann integral is not defined for a wide range of functions and situations of importance in applications (and of interest in theory). For example, the Riemann integral can easily integrate density to find the mass of a steel beam, but cannot accommodate a steel ball resting on it. This motivates other definitions, under which a broader assortment of functions are integrable. The Lebesgue integral, in particular, achieves great flexibility by directing attention to the weights in the weighted sum...*

Kind regards

Re: \int_0^{\infty}\frac{\sin x}{x}\ dx

Also sprach Zarathustra, have you tried double integration? It's a neat approach and the justification is easy.

Re: \int_0^{\infty}\frac{\sin x}{x}\ dx

Quote:

Originally Posted by

**Krizalid** Also sprach Zarathustra, have you tried double integration? It's a neat approach and the justification is easy.

No, I didn't.

I *just* want to understand that interesting method that the author presenting.

By the way, *he* using in the computation, the interesting formula for ,

.