How to do intgral for (cos(x)^2)*cos(wx)

[mirshayan]

Homework Statement

Homework Equations

The Attempt at a Solution

 

[mirshayan]

excuse me
i want integral of cos(x^2)*cos(wx)dx

 

[Samy_A]

excuse me
i want integral of cos(x^2)*cos(wx)dx

I "want" that too.

1. Are you sure that is the exercise? No typo?
2. Please show your attempt(s) to solve the integral.

 

[mirshayan]

I "want" that too.

1. Are you sure that is the exercise? No typo?
2. Please show your attempt(s) to solve the integral.

actually its the Fourier cos transforms of cos(x^2)
our teacher ask us To prove it. i know the answer is what (see the attach file) but i want way

 

[Samy_A]

There are probably a number of ways to prove it.

I would start by noting that $\cos (y)=\frac{e^{iy}+e^{-iy}}{2}$.

But show us how you would try to find the answer, please.
Remember that the real question is not to find the integral of $\cos(x²)\cos(\omega x)$, but to find the Fourier transform of $\cos(ax²)$. You could at least start by defining that Fourier transform in the "relevant equations" part of the template.

 

[mirshayan]

thx a lot

There are probably a number of ways to prove it.

I would start by noting that $\cos (y)=\frac{e^{iy}+e^{-iy}}{2}$.

But show us how you would try to find the answer, please.
Remember that the real question is not to find the integral of $\cos(x²)\cos(\omega x)$, but to find the Fourier transform of $\cos(ax²)$. You could at least start by defining that Fourier transform in the "relevant equations" part of the template.