Another fun integral

**Mathstud28** · July 18th 2008, 07:03 PM

$\int_0^{\infty}\bigg[\frac{\sin^2(px)\cos^2(px)}{x^2}-\frac{\sin^4(px)}{3x^2}\bigg]~dx$

**Chris L T521** · July 18th 2008, 07:14 PM

Originally Posted by Mathstud28

$\int_0^{\infty}\bigg[\frac{\sin^2(px)\cos^2(px)}{x^2}-\frac{\sin^4(px)}{3x^2}\bigg]~dx$

I got to this point...and then I'm kinda stuck...any hints?

$\int_0^{\infty}\frac{\sin^2(px)}{x^2}\,dx-\frac{4}{3}\int_0^{\infty}\frac{\sin^4(px)}{x^2}\, dx$

--Chris

**Mathstud28** · July 18th 2008, 07:19 PM

Originally Posted by Chris L T521

I got to this point...and then I'm kinda stuck...any hints?

$\int_0^{\infty}\frac{\sin^2(px)}{x^2}\,dx-\frac{4}{3}\int_0^{\infty}\frac{\sin^4(px)}{x^2}\, dx$

--Chris

Going that route?

$\frac{1}{x^2}=\int_0^{\infty}ye^{-yx}~dy$

**Mathstud28** · July 18th 2008, 07:33 PM

Originally Posted by Chris L T521

I got to this point...and then I'm kinda stuck...any hints?

$\int_0^{\infty}\frac{\sin^2(px)}{x^2}\,dx-\frac{4}{3}\int_0^{\infty}\frac{\sin^4(px)}{x^2}\, dx$

--Chris

I didn't just edit my quote so you would see this, I went through the calculations of this and I think it is wrong

**Mathstud28** · July 19th 2008, 03:02 PM

Hint:

Let $\int_0^{\infty}\bigg[\frac{\sin^2(px)\cos^2(px)}{x^2}-\frac{\sin^4(px)}{3x^2}\bigg]~dx=f(p)$

**Mathstud28** · July 21st 2008, 02:05 PM

Originally Posted by Mathstud28

$\int_0^{\infty}\bigg[\frac{\sin^2(px)\cos^2(px)}{x^2}-\frac{\sin^4(px)}{3x^2}\bigg]~dx$

No one's going to bite?

This is

$\frac{1}{12}\frac{d^2}{dp^2}\int_0^{\infty}\frac{\ sin^4(px)}{x^4}~dx=\frac{\pi{p}}{6}$

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