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

In summary, the conversation is about finding the Fourier transform of cos(ax²) and proving it using various methods. The initial request was for the integral of cos(x^2)*cos(wx)dx, but it was clarified that the actual question is about the Fourier transform. The conversation ends with a reminder to define the Fourier transform in the relevant equations section of the template.
  • #1
mirshayan
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Member warned about posting with no effort shown

Homework Statement

Homework Equations

The Attempt at a Solution

 
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  • #2
excuse me
i want integral of cos(x^2)*cos(wx)dx
 
  • #3
mirshayan said:
excuse me
i want integral of cos(x^2)*cos(wx)dx
I "want" that too. :smile:

1. Are you sure that is the exercise? No typo?
2. Please show your attempt(s) to solve the integral.
 
  • #4
Samy_A said:
I "want" that too. :smile:

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.:headbang::headbang::headbang: i know the answer is what (see the attach file) but i want way
 

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  • #5
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.
 
  • #6
thx alot
Samy_A said:
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.
 

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

1. What is the formula for integrating (cos(x)^2)*cos(wx)?

The formula for integrating (cos(x)^2)*cos(wx) is ∫ cos(x)^2 * cos(wx) dx = (1/2) * cos(2x) + (1/2w) * sin(2wx) + C.

2. What is the step-by-step process for integrating (cos(x)^2)*cos(wx)?

The step-by-step process for integrating (cos(x)^2)*cos(wx) is as follows:

  1. Use the trigonometric identity cos^2(x) = (1 + cos(2x))/2 to rewrite the integral as ∫ (1 + cos(2x))/2 * cos(wx) dx.
  2. Distribute the 1/2 into the integral to get ∫ (1/2) * cos(wx) dx + ∫ (1/2) * cos(2x) * cos(wx) dx.
  3. Integrate the first integral using the substitution u = wx, du = w dx to get (1/2w) * sin(wx) + C.
  4. Use the trigonometric identity cos(a) * cos(b) = (1/2) * (cos(a+b) + cos(a-b)) to rewrite the second integral as ∫ (1/4) * (cos(3wx) + cos(wx)) dx.
  5. Integrate the second integral using the substitution u = 3wx, du = 3w dx to get (1/6w) * sin(3wx) + (1/4w) * sin(wx) + C.
  6. Combine the results from step 3 and 5 to get the final solution: (1/2) * cos(2x) + (1/2w) * sin(2wx) + C.

3. Can the integral (cos(x)^2)*cos(wx) be solved using any other methods?

Yes, the integral (cos(x)^2)*cos(wx) can also be solved using the method of integration by parts. However, the steps and final solution will be the same.

4. How do you handle the constant w when integrating (cos(x)^2)*cos(wx)?

The constant w can be treated as a constant throughout the integration process. It only affects the final result by multiplying the sine term by 1/w.

5. Can the integral (cos(x)^2)*cos(wx) be solved for any value of w?

Yes, the integral (cos(x)^2)*cos(wx) can be solved for any value of w. However, the steps and final solution may vary depending on the value of w.

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