How to integrate Acos(wt + theta) ?

Then du= \omega dt. So \int cos(\omega t+ \theta) dt= \int cos(u) \omega^{-1} du, which you can now integrate.In summary, the conversation discusses how to integrate Acos(\omegat + \theta) with A being the amplitude, omega being the angular velocity, and theta being the position. It is suggested to use the substitution method and let u = \omega t + \theta to simplify the integral. The conversation also mentions that if T represents the period, then the integral may not need to be evaluated as there is no DC content in a sinusoid.
  • #1
Chandasouk
165
0
How do you integrate Acos([tex]\omega[/tex]t + [tex]\theta[/tex]) ? Where A is the amplitude, omega is angular velocity, and theta is position? I have no idea what to do. Should I U substitute?
 
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  • #2
With respect to theta? omega?
 
  • #3
I'm actually not sure, the integral is being taken from 0 to T and the equation is Acos([tex]\omega[/tex]t + [tex]\theta[/tex])dt
 
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  • #4
Yes, you can U substitute.
Or ask yourself what is the derivative of sin(wt+theta).

If the capital T is period(as usual), you don't need to actually integrate it and write down 0 as the answer. Because there is no DC content in a sinusoid.
 
  • #5
Chandasouk said:
I'm actually not sure, the integral is being taken from 0 to T and the equation is Acos([tex]\omega[/tex]t + [tex]\theta[/tex])dt

That dt tells you that integration is to be done with respect to t, so as far as the integration is concerned, t is the variable and the other two are just constants.
 
  • #6
You would be able to integrate [itex]\int cos(x) dx[/itex] wouldn't you? So it is just that [itex]\omega t+ \theta[/itex] that is the problem.

So let [itex]u= \omega t+ \theta[/itex].
 

FAQ: How to integrate Acos(wt + theta) ?

What is the formula for integrating Acos(wt + theta)?

The formula for integrating Acos(wt + theta) is ∫Acos(wt + theta) dt = (A/w)sin(wt + theta) + C, where A is the amplitude, w is the frequency, and C is the constant of integration.

How do I determine the limits of integration for Acos(wt + theta)?

The limits of integration for Acos(wt + theta) depend on the given problem. They can be determined by considering the starting and ending points of the function or by using the given time period.

3. Is it possible to integrate Acos(wt + theta) without using trigonometric identities?

Yes, it is possible to integrate Acos(wt + theta) without using trigonometric identities by using the substitution method. Let u = wt + theta, then du = wdt. The integral becomes ∫Acos(u) du = Asin(u) + C = Asin(wt + theta) + C.

4. Can I use the power rule to integrate Acos(wt + theta)?

No, the power rule cannot be used to integrate Acos(wt + theta) as it is not a polynomial function. It is a trigonometric function and requires the use of trigonometric identities or the substitution method for integration.

5. How do I solve for the constant of integration in the integration of Acos(wt + theta)?

The constant of integration, C, can be solved for by using the given initial conditions or by evaluating the integral at a specific point. For example, if the initial condition is f(0) = 2, then the constant of integration can be found by setting t = 0 and solving for C.

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