Solving Integral of sinc(x) for Definite Integral

[Sistine]

Homework Statement

I'm trying to prove the following definite integral of $$sinc(x)$$

$$\int_{-\infty}^{\infty}\frac{\sin(x)}{x}dx=\pi$$

Homework Equations

The Attempt at a Solution

I've tried power series expansions as well as trigonometric identities like

$$\frac{\cos 2x}{x}=\frac{\cos^2 x}{x}-\frac{\sin^2 x}{x}$$

I also looked at techniques used to integrate the definite integral
$$\int_{-\infty}^{\infty}e^{-x^2}dx$$

which I know is solved by double integration and changing to polar coordinates. However, this does not help me integrate $$sinc(x)$$.

 

[phsopher]

Well, I suppose you could do it by making a closed curve in the complex plane and using Caychy's theorem (and Jordan's lemma). There might be an easier way, but I can't think of any.

 

[elliotician]

Think about euler's formula and leibniz. A 'simple' proof can be made this way.

 

[Count Iblis]

$$\int_{0}^{\infty}\sin(x)\exp(-sx)dx=\frac{1}{1+s^2}$$

Integrate both sides from s = 0 to infinity to obtain the result.

 

[elliotician]

Check out this proof of the dirichlet integral:

http://en.wikipedia.org/wiki/Dirichlet_integral

 

[ahmedxahmed]

sinc(x) = sin(x)
x
has no anti-derivative

 

[Whitishcube]

sinc(x) = sin(x)
x
has no anti-derivative

no elementary anti-derivative