# How to calculate this integral?

1. May 11, 2006

### neptunecs

Hi,How to calculate this integral:

which
I have ever calculated it and the result(R(s)) is ∞,but I don't think it's right,I'm not sure with it.
neptunecs.

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2. May 11, 2006

### Curious3141

It'll take forever for the attachments to be approved, why not just use LaTex?

3. May 11, 2006

### neptunecs

where is LaTex in this forum?I didn't find it.Tell me,please.Thanks.

4. May 11, 2006

### Curious3141

5. May 11, 2006

### neptunecs

R(s)=$$\int_{-\infty}^{\infty} f(t)f(t+s) dt$$
which f(t)=Acos(wt+f0)
Thank for Curious3141's help!
neptunecs.

Last edited: May 11, 2006
6. May 11, 2006

### J77

$$\int\cos mx \cos nx dx=\frac{\sin(m-n)x}{2(m-n)}+\frac{\sin(m+n)x}{2(m+n)}+c$$

Of course the c will go because you have limits.

(This integral can be found in the front or back of most Calculus textbooks.)

7. May 11, 2006

### Curious3141

You're welcome. If you're not that interested in the method, a quick way to get an answer is to try this link : http://integrals.wolfram.com/index.jsp

Just input the integrand as "b Cos[w x+k]b Cos[w(x+l)+k]" since the script uses x as the integrand by default. The constant I changed from A to b because Acos has a special meaning of arccosine. f0 corresponds to k and s corresponds to l.

EDIT : I had previously made a small error in the expression which changes the indef. integral but does not affect the convergence of the integral for the given bounds. Now the integrand expression is correct.

Last edited: May 11, 2006
8. May 11, 2006

### Curious3141

And you're right, for the bounds given, the integral does not converge.

9. May 11, 2006

### Pseudo Statistic

Or he could've converted the integrand to complex form.. :D

10. May 11, 2006

### neptunecs

Thank you.
But the integral does not converge seems strange.Maybe my teacher made a mistake.

11. May 12, 2006

### J77

I made a mistake in giving you that standard solution due to the phase shifts in your integrands...

Using the mathematica link, the solution should be:

$$\frac{A^2}{2}\left[t\cos(\omega s)+\frac{\cos(2\omega t)\sin(2f_0+\omega s)}{2\omega}+\frac{\cos(2f_0+\omega s)\sin(2\omega t)}{2\omega}\right]^\infty_{-\infty}$$

Though you should check it too... the non-convergence thing still stands...

12. May 13, 2006

### neptunecs

Thank very much.This solution is particular.
my teacher said It's non-convergence thing, too.
neptunecs.