Difficult integral

[perishingtardi]

Mod note: Moved from the math technical sections.
I need to show that
$$\int_{-\infty}^\infty \frac{\sin^2 (pa/\hbar)}{p^2} \, dp = \frac{\pi a}{\hbar}.$$
I haven't got a clue how to integrate this function! Any help would be much appreciated thanks.

 

[perishingtardi]

I've found through the transformation $u=pa/\hbar$ that it is equivalent to showing
$$\int_{-\infty}^{\infty} \frac{\sin^2 u}{u^2}\,du = \pi,$$ if that helps anyone.

 

[SteamKing]

You've got your substitution and your integral.

All you have to do is figure out p and dp in terms of u and du.

The algebra is really simple.

 

[perishingtardi]

You've got your substitution and your integral.

All you have to do is figure out p and dp in terms of u and du.

The algebra is really simple.

Yeah I did that substitution myself to make it easier... now how do I show that the integral is equal to pi??

 

[voko]

Using the cosine of double angle formula, and then integration by parts, this can be reduced to integrals of $$ \frac {\sin x} {x} $$

 

[amiras]

Under a change of variable x = pa/h:

$$\frac{\hbar}{a}\int_{-\infty}^\infty \frac{sin^2 x}{x^2}dx$$

As suggested you can apply integration by parts and then sine double angle rule to obtain

$$\frac{\hbar}{a}\int_{-\infty}^\infty \frac{sin(2x)}{x}dx$$

The integral below has many different proofs, few of them are here
$$\int_{-\infty}^\infty \frac{sin(ωx)}{x}dx=\pi$$ for any ω > 0.

 

[Crake]

OFF-TOPIC:

Can someone tell me what is the difference between a question asked here and a question asked on the homework forums?

This is an honest question. For example, shouldn't this question ("Difficult integral") be in the homework section?

Sorry for the off topic perishingtardi

 

[Mark44]

Crake,
You are correct. As the sticky says at the top of this forum section, "This forum is not for homework or any textbook-style questions."

I am moving this thread to the Homework section.