Definite integral ∫(cos4x−cos4α)/(cosx−cosα)dx

In summary, the definite integral is used to find the area under the curve of a function, specifically between the curves of cos4x and cos4α divided by the difference between cosx and cosα. It is possible to find the exact value of the definite integral using various methods, but it may result in a complex expression. The values of α affect the definite integral by changing the limits of integration, resulting in a larger or smaller area under the curve. This definite integral can also be solved numerically using methods such as the trapezoidal rule or Simpson's rule. Additionally, it has real-life applications in fields such as physics, engineering, and economics.
  • #1
lfdahl
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Evaluate the definite integral:\[I = \int_{0}^{\pi}\frac{\cos 4x - \cos 4\alpha }{\cos x - \cos \alpha }dx\]- for some $\alpha \in \mathbb{R}.$
 
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  • #2

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  • #3
[sp]It looks as though the answer should be $I = 4\pi\cos2\alpha\cos\alpha$ (because the constant is integrated over an interval of length $\pi$). (Cool)
[/sp]
 
  • #4
Opalg said:
[sp]It looks as though the answer should be $I = 4\pi\cos2\alpha\cos\alpha$ (because the constant is integrated over an interval of length $\pi$). (Cool)
[/sp]

Yes, indeed. A factor $\pi$ is missing in the answer. I am so sorry for this typo!
 

FAQ: Definite integral ∫(cos4x−cos4α)/(cosx−cosα)dx

What is the purpose of calculating the definite integral of this function?

The definite integral is used to find the area under the curve of a function. In this case, it is used to find the area between the curves of cos4x and cos4α divided by the difference between cosx and cosα.

Is it possible to find the exact value of the definite integral?

It is possible to find the exact value of the definite integral using various methods such as the substitution method or integration by parts. However, it may result in a complex expression.

How do the values of α affect the definite integral?

The values of α affect the definite integral by changing the limits of integration. As α increases, the limits of integration also increase, resulting in a larger area under the curve. Similarly, as α decreases, the limits decrease, resulting in a smaller area.

Can this definite integral be solved numerically?

Yes, this definite integral can be solved numerically using methods such as the trapezoidal rule or Simpson's rule. These methods approximate the area under the curve by dividing it into smaller trapezoids or parabolas, respectively.

Are there any real-life applications of this definite integral?

Yes, this definite integral can be used in various fields such as physics, engineering, and economics to calculate the work done, energy consumed, or cost incurred over a specific time period.

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