Q: Integration of e^(-t/2)cos(n2t)dt

  • Thread starter Peter_L
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In summary, the person is struggling to integrate the function e^(-t/2)cos(n2t)dt and is seeking help from others. They suggest using the u substitution technique or integration by parts, but have not been able to find a suitable substitute for u. The limits for integration are from 0 to pi.
  • #1
Peter_L
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Hello everyone,

I haven't done calculus in a long time now and I've been reviewing some old calc books but couldnt' find anything that helps. Was wondering how to integrate this function:

e^(-t/2)cos(n2t)dt

I was thinking of using the u substitution technique but can't find a substitute for u. Any help would be fantastic.

Thanks.
 
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  • #2
Limits are from 0 to pi

Limits are from 0 to pi
 
  • #3
well from what i can see here, i haven't tried it myself though, so i might be wrong, but it is worth a try. integration by parts i think would work here. let u=e^-(t/2), and v=integ of cos(n2t)dt, and applying integration by parts twice i think you might get something.
sorry for not having time to do it myself first.
 

1. What is the purpose of integrating e^(-t/2)cos(n2t)dt?

The purpose of integrating e^(-t/2)cos(n2t)dt is to find the value of the function over a given interval. Integration is a mathematical process that allows us to calculate the area under a curve, which can be useful in various fields such as physics, engineering, and economics.

2. What is the general formula for integrating e^(-t/2)cos(n2t)dt?

The general formula for integrating e^(-t/2)cos(n2t)dt is ∫e^(-t/2)cos(n2t)dt = (e^(-t/2) / (1+(n2)^2)) * (cos(n2t) + n2sin(n2t)) + C, where C is the constant of integration.

3. How do you solve the integral of e^(-t/2)cos(n2t)dt?

To solve the integral of e^(-t/2)cos(n2t)dt, you can use integration by parts or the substitution method. Both methods involve breaking down the integral into simpler parts and then using known integration rules to solve it. Alternatively, you can also use a computer software or calculator to find the numerical value of the integral.

4. What is the range of values for n in the integral of e^(-t/2)cos(n2t)dt?

The range of values for n in the integral of e^(-t/2)cos(n2t)dt is all real numbers except for 0. This is because when n=0, the integral becomes ∫e^(-t/2)dt, which is a straightforward integration problem with a solution of -2e^(-t/2) + C. For all other values of n, the integral can be solved using the general formula mentioned in question 2.

5. What are some real-life applications of integrating e^(-t/2)cos(n2t)dt?

The integral of e^(-t/2)cos(n2t)dt has many real-life applications, such as in physics to calculate the displacement, velocity, and acceleration of an object under the influence of a decaying force. It is also used in engineering to analyze the response of a system to a damped harmonic force. Moreover, it can be used in economics to calculate the present value of a series of future cash flows.

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