# Integral of sinc(x)

1. Jun 1, 2009

### Sistine

1. The problem statement, all variables and given/known data
I'm trying to prove the following definite integral of $$sinc(x)$$

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

2. Relevant equations

3. 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 $$sinc(x)$$.

2. Jun 1, 2009

### 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.

3. Jun 1, 2009

### elliotician

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

4. Jun 1, 2009

### 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.

5. Jun 2, 2009

### elliotician

6. Jan 19, 2011

### ahmedxahmed

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

7. Jan 19, 2011

### Whitishcube

no elementary anti-derivative