# Thread: Stumped on an integral

1. ## Stumped on an integral

Working on a Fourier cosine transform and need a solution to the following integral to continue. Very grateful for any responses/ advice. My integration is a little rusty.

Integral ((Sin 2pix Cos piX) / X)

-Dave

2. Originally Posted by DMessin
Working on a Fourier cosine transform and need a solution to the following integral to continue. Very grateful for any responses/ advice. My integration is a little rusty.

Integral ((Sin 2pix Cos piX) / X)

-Dave
What are the bounds of the integral? If it is this

$\displaystyle \int_0^{\infty}\frac{\sin(2\pi{x})\cos(\pi{x})}{x} dx$

The answer is $\displaystyle \frac{\pi}{2}$

To see this rewrite your integral as

\displaystyle \begin{aligned}\int_0^{\infty}\frac{\sin(2\pi{x})\ cos(\pi{x})}{x}dx&=\int_0^{\infty}\sin(2\pi{x})\co s(\pi{x})\int_0^{\infty}e^{-yx}dydx\\ &=\int_0^{\infty}\int_0^{\infty}e^{-yx}\cos(\pi{x})\sin(2\pi{x})dydx \end{aligned}

And you may switch the integration order because it is over a rectangular region by Fubini's Theorem.

3. ## THANK YOU!!!!

Brilliant, thanks!!!!!!!