# Difficult integral

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.

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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
Staff Emeritus
Homework Helper

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

The algebra is really simple.

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??

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

1 person
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.

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
Mentor
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.